Diboson at the LHC vs LEP

We use the current CMS and ATLAS data for the leptonic $pp \to WW, WZ$ channels to show that diboson production is, for a broad class of flavour models, already competitive with LEP-1 measurements for setting bounds on the dimension six operators parametrising the anomalous couplings between the quarks and the electroweak gauge bosons, at least under the assumption that any new particle is heavier than a few TeV. We also make an estimate of the HL-LHC reach with $3$ ab$^{-1}$. We comment on possible BSM interpretations of the bounds, and show the interplay with other searches for a simplified model with vector triplets. We further study the effect of modified $Z$-quark-quark couplings on the anomalous triple gauge coupling bounds. We find that their impact is already significant and that it could modify the constraints on $\delta g_{1z}$ and $\delta \kappa_\gamma$ by as much as a factor two at the end of HL-LHC ($\lambda_\gamma$ is only marginally affected), requiring a global fit to extract robust bounds. We stress the role of flavour assumptions and study explicitly flavour universal and minimal flavour violation scenarios, illustrating the differences with results obtained for universal theories.


Introduction
The Large Hadron Collider (LHC) is probing the Standard Model (SM) at higher energies than ever before, reaching new regions never explored so far. For this reason, we must take the chance to learn as much as possible from it. With the discovery of a scalar particle consistent with the Higgs boson, the SM can in principle be consistent up to the Planck scale. Nonetheless in many UV completions predicting a light Higgs, e.g. supersymmetric or composite Higgs models, one requires other new particles with masses around the electroweak scale unless there is some fine tuning. So far though, the LHC has not seen any robust hints of new physics, which indicates that any new particles must be either too weakly coupled to the SM or heavy enough to not have been seen. In the first case, one may expect to see direct effects, like for example a resonance showing up once enough luminosity is collected. In the second case, one expects to see the effects of the new particles indirectly, for example, by modifying the differential cross sections of particular processes with respect to the SM prediction.
The study of diboson production, pp → W V, V = W, Z, offers a way to probe physics scenarios of the second class. The interest in such channels both at lepton and hadron colliders is not new [1][2][3] but it has recently received renewed attention, see in particular Refs. [4][5][6][7][8][9][10][11][12][13][14]. This is first due to the fact that together with pp → V h [15,16], diboson production directly probes the interactions of the Goldstone bosons via the gauge boson longitudinal polarizations, and therefore is one of the first places where to expect signs of new physics related to the electroweak symmetry breaking. Furthermore, in Refs. [6,10,12] it has been shown that at high energy, the leading amplitudes for pp → V V, V h grow with the center of mass energy faster than the SM ones and therefore diboson production can benefit from the higher energy probed at the LHC to reveal sign of new physics. See Refs. [9,12,15,[17][18][19][20][21][22][23] for studies using this high energy behaviour to increase the sensitivity to d = 6 operators; notice that in some cases the new LHC bounds can improve on the LEP-1 and LEP-2 bounds. As shown in Refs. [6,10,12,24], in the Higgs basis and at the dimension-six level [25,26], there are a priori seven Beyond the Standard Model (BSM) coefficients that modify the diboson amplitude pp → W V at high energies. These are three anomalous triple gauge couplings (aTGC), traditionally parametrised by δκ γ , δg 1z , λ γ , and four anomalous couplings between the light quarks and the Z gauge boson, δg Zu L , δg Zu R , δg Zd L , δg Zd R (δVqq hereinafter), that will be introduced later. Interestingly, the pp → V V and pp → V h amplitudes at high energy become equal, as expected by the Goldstone equivalence theorem, and actually only depend on five combinations of the d = 6 operators [6,10,12]. There has not been yet a complete global analysis establishing the future bounds on these five independent so-called High Energy Parameters, but some first results have been obtained in the W Z [12] and Zh [16] channels, showing some nice complementarity. Combining with LEP constraints on the Z couplings to fermions, one could in principle univocally derive bounds on the aTGCs. The purpose of our work is to stress that, if this strategy is perfectly fine for universal theories, the aTGC bounds obtained that way do not directly apply when other flavour assumptions are considered and one needs to perform a global fit to derive bounds on both aTGCs and Vqq couplings.
In this work, we study the constraining power of diboson data to set bounds on the anomalous couplings between the W and Z gauge bosons and the light quarks. In particular, we make use of the differential distributions reported by the experimental collaborations. We find that due to the enhanced sensitivity at high energies, pp → W V can already be competitive or even surpass LEP-1 on setting bounds on δVqq, at least under the assumptions that these anomalous couplings are generated by new particles with masses equal or greater than a few TeV to ensure the validity of the EFT, see section 5.1. We refer the reader to Refs. [4,9,11,12,23] where new differential distributions and experimental searches are proposed in order to increase the sensitivity to the effective field theory (EFT) operators entering diboson production. If these are implemented by the experiments, the increase of sensitivity could allow diboson production at HL-LHC to set much stringent bounds on the BSM amplitudes, reaching the point where they are smaller than the SM, and therefore can start constraining BSM scenarios with a characteristic coupling smaller than a typical SM gauge coupling.
We rely on the differential distributions from the up-to-date diboson measurements performed by ATLAS and CMS with up to 20 fb −1 of data at 8 TeV and 13 fb −1 of data at 13 TeV, see Table 1 for details. We also estimate the sensitivity expected at the high-luminosity run of the LHC (HL-LHC) with an anticipated total of 3 ab −1 of data. We consider two general flavour structures of the higher dimensional operators: i) Flavour Universality (FU), where the EFT operators satisfy a U (3) 5  δ ij . 1 It would also be interesting to combine and compare the LEP-1 bounds on the δVqq couplings with other flavour scenarios, e.g. the anarchic case (see Ref. [27]) or the diagonal one, i.e. diag(δg Zu,d L,R 11 , δg Zu,d L, R 22 , δg Zu,d L,R 33 ). We leave these analysis for future work since the non-Gaussianity of the fit makes it non-trivial to go from a more general case to a more restrictive one. Diboson production at hadron colliders is insensitive to these assumptions since the cross section is dominated by the light quarks, while the constrains from LEP-1 can change by an order of magnitude, see the results of Ref. [27] that we summarise in Appendix B. Another interesting UV assumption is that of universal theories [28][29][30]. These can be defined as those theories whose EFT can be fully described by bosonic operators and deviations of the light quark couplings can be written in terms of the gauge boson oblique parameters. Given that the LEP-1 bounds for these types of theories is one or two orders of magnitude stronger than those for MFV and FU, we found that with the current experimental searches diboson production is not competitive with LEP-1 for unviersal theories. 2 1 When simulating the diboson production, we only modify the couplings to the u and d quark since the BSM effects from the heavier quarks are PDF-negligible, below 1%, within the flavour assumptions considered. Therefore the diboson analysis presented in this paper does not distinguish MVF and U (2) 5 -flavour symmetric setups. 2 See Appendix B for the bounds on universal theories expressed in the Higgs basis.
Due to the larger systematics at the LHC, the conclusion that it can surpass LEP-1 and LEP-2 in setting bounds on the EFT operators may come as a surprise, but it follows from the fact that some BSM amplitudes can grow with the characteristic scale of the hard process probed at the LHC. However, it is important to keep in mind, as stressed in Refs. [12,15,17,20,21], that the larger systematics also imply in many cases that the new LHC bounds are valid only when the BSM contribution is larger than the SM one, limiting in some cases the generality of these bounds to subsets of possible UV theories. We comment more on the EFT interpretation in section 5. Nonetheless, given that the LHC is running and we do not know what new physics may lie ahead, it is still important to make sure that all the regions of the EFT parameter space are explored in the most model independent way as possible.
Besides studying the bounds on δVqq, we also look at the impact of non-vanishing δVqq in the aTGC determination under the different flavour assumptions considered. Looking at this effect was first mentioned and motivated in Refs. [7,31] and checked explicitly in Ref. [10] using the channel pp → W W at 8 TeV by ATLAS. We extend this analysis by first performing a global fit to the present data for all the channels in Table 1 and also by studying the impact of different flavour assumptions. We also estimate the sensitivity that one can hope to reach at HL-LHC, concluding that the effect of δVqq will be more and more important in the future. It should be noted that the analysis we provide is done at leading order (LO). We expect that the NLO effects are most relevant for amplitudes with final transverse polarizations due to the non-interference effects shown in Ref. [10,32]. Since in our study we are mostly interested in the longitudinal polarizations, we do not expect much difference in our conclusions even though it would be interesting to study in more detail the NLO effects.
We briefly comment on possible interpretations of the EFT bounds derived. To gain perspective and a sense of the usefulness of the constraints coming from diboson production, we study a simplified model of heavy vector triplets and compare the diboson bounds to the ones from other searches like dijets, resonant diboson or Higgs coupling measurements, finding that diboson can be complementary to other processes in exploring the parameter space of the model, and can be the leading probe in important regions of parameters.
The paper is organized as follows. In section 2, we give the conventions and review the high energy behavior of diboson production at the LHC. In particular, we note that the highenergy diboson amplitudes in the Higgs basis are controlled by seven independent parameters in FU and MFV setups as opposed to five parameters only for universal theories. In section 3, we present the bounds on the δVqq and the effect of allowing these to be non-zero in the aTGC exclusion plots. In section 4, we estimate the δVqq bounds that can be expected by the end of HL-LHC and we quantify the effect of letting δVqq and aTGC float in global EFT fit. In section 5, we briefly review the validity of the EFT approach in presence of non-negligible contributions from the dimension six BSM quadratic amplitudes, and we review various UV scenarios and power counting rules which motivate the various assumptions on the values of the parameters used through the paper. We also study a toy model with heavy triplets as a concrete example. And we compare our HL-LHC bounds for this toy model with those coming from Higgs coupling measurements and dijet searches. We conclude in section 6. Four appendices provide further technical details and cross-checks.

Theoretical framework
We work in the so called Higgs basis [24][25][26], and follow the conventions of Ref. [25] where α, G F and m Z are taken as the input parameters. The Higgs basis parametrizes the d = 6 EFT operators as modifications to the SM vertices, and where the fields are in the mass eigenstates and in the unitary gauge. In this basis and considering only operators with d ≤ 6, the relevant terms for pp → W V production are: The first term contains the SM interactions between the electroweak gauge bosons together with the d = 6 aTGC deformations, The second term in the Lagrangian (1) contains the SM contribution and deviations to the couplings between the up and down quarks to the W, Z, gauge bosons, Since at dimension six the following relations are satisfied (see for instance Ref. [25]): the deviations of the schematic form ∼ g SM (1 + δ) can be parametrized by two independent aTGC (which we choose to be δκ γ , δg 1z ), and four independent corrections to Zqq vertices (which we choose to be δg Zu L , δg Zu R , δg Zd L , δg Zd R ). Notice that the aTGC parametrized by λ γ = λ z introduces a new type of coupling non-existent in the SM. In total there are, in the Higgs basis, seven parameters that contribute to the leading deformations to diboson production (three aTGC and four δVqq).
In the Lagrangian (3) we have not included right-handed charged currents nor dipole contributions since under FU and MFV they are either zero, or are suppressed by the Yukawas of the light quarks. We also ignore the deformations in the lepton sector since their bounds from LEP-1 data are an order of magnitude better than those on the quark sector [27]. Finally we also ignored the shift to the W mass, δ m , since its current existing bound is such that it numerically gives in diboson production an effect ten times smaller than a modified quark coupling.

High energy behaviour and correlations
In the Higgs basis, the energy growth of the amplitudes that interfere with the SM in the high energy limit can be understood as follows. At tree level in the SM, the leading amplitude for qq → W W (W Z) is given by the sum of three diagrams, consisting of an s-channel exchange of the γ, Z bosons (W boson), and a t-channel contribution. Taking as an example the case ofqq → W W , whereqq =ūu,dd, one finds that the tree level SM amplitude is given by the Feynman diagrams of Fig. 1. One can check that at large center of mass energy,ŝ m 2 W , the total amplitude for qq → W + 0 W − 0 is given by [33] where W ± 0 stand for the longitudinal polarizations of the W ± gauge bosons,ŝ is the squared center of mass energy, the dots denote sub-leading contributions at high energy, Q q and T 3 q are the electric charge and SU (2) weak isospin of the initial quarks and θ is the angle between W + and the beam axis. 3 The key point of Eq. (5) is to notice that while each of the individual sub-amplitudes grows withŝ, the sum does not. Therefore, any shift to the SM couplings, shown in blue and red in Fig. 1, will spoil the cancellation of the different pieces in Eq. (5), and therefore the resulting amplitude will be proportional toŝ. In the Higgs basis it is especially clear to see that all the coefficients modifying diboson production with a shift to the SM couplings will generically induce an amplitude that grows withŝ. Figure 1: Representative contributions to diboson production. The sensitivity of the measurements, already with O(20) fb −1 of data and certainly even more at HL-LHC, is such that they can improve the LEP-1 constraints on the quark couplings to gauge bosons (blue). This also implies that the LEP-1 bounds are no longer stringent enough to make these parameters negligible when setting bounds on the anomalous triple gauge couplings (red).
Notice that the interaction given by (3) is not present in the SM. In this case one cannot use the spoiling of the SM amplitude cancellation of Eq. (5) to see whether its effect asymptotically grows with the center of mass energy. Nonetheless, one can see by direct calculation that the amplitude induced by this operator actually grows withŝ as a consequence of the presence of extra derivatives in the interaction.

Helicity amplitudes at high energy and correlations between aTGC and δVqq
To estimate which operators or combinations of operators will be the most constrained by diboson production at the LHC, one can study each helicity amplitude as done in Refs. [6,12].
In the limit whereŝ m 2 W , the leading helicity amplitudes for the partonic scattering qq → W W are given by 4 where δg W q L = δg Zu L − δg Zd L and δg Zq L,R corresponds to the anomalous vertex of the incoming quark q, defined in Eq. (3). For qq → W Z, the energy growing amplitudes are We can see, as pointed out in Refs. [6,12], that in the asymptotic high energy regime there are only five independent combinations of parameters entering pp → W V since Therefore, there are only four relevant independent combinations for the longitudinal polarizations and one for the transverse ones that can be probed in the high energy limit. 5 These four directions for the longitudinal polarizations are the so-called High Energy Parameters (HEPs) introduced in Ref. [12], see Table 2 in this reference. For completeness, in Appendix D, Eq. (49), these four HEPs are written explicitly in terms of the Higgs basis [25] parameters.
In Appendix E we express the amplitudes shown in Eqs. (6), (7) in the Warsaw basis. Notice that if the experimental sensitivity is low such that the quadratic BSM squared amplitudes dominate the cross section, all the channels above show a similar behaviour at 4 The L, R stand for the initial helicities of the quarks, while ± and 0 stand for the transverse and longitudinal polarizations of the final electroweak bosons respectively. We computed these amplitudes using FeynCalc [34] using the BSMC package [35] for FeynRules [36], finding agreement with the expressions presented in Ref. [6], which also was a cross check for the .ufo file used in the Madgraph5 simulations. 5 This counting may change for other flavour assumptions, since the right-handed charged current, that could be present away from the FU/MFV setups, gives rise also to an energy-growing amplitude for pp → W Z: M(RR; 00) = −i e 2ŝ sin θ high-energy. On the other hand, when the experimental sensitivity is getting good enough to probe BSM deformations subdominant to the SM, the channels that feature an interference between SM and BSM will be of better use to bound anomalous couplings. As shown in Refs. [12,32], this selects the production of two longitudinally gauge bosons as the preferred channel. It is nonetheless possible to also use the production of transversally polarized gauge bosons when relying on specific kinematic observables to resurrect the interference [9,11]. In our analysis, which uses the current experimental observables, the quadratic pieces are always at least a factor ten larger than the interference parts, and therefore the question of the BSM/SM interference is not relevant in our analysis.
From Eqs. (6) and (7), there are a total of seven coefficients parameterizing the five directions growing asŝ in the processes pp → W V . Hence, in the asymptotic high energy limit, two completely flat directions are anticipated among the Higgs basis coefficients. A simple way to see the flat directions explicitly is by noting that any deviation of δg 1z and δκ γ in Eqs. (6), (7) can be compensated by a modification of the vertex corrections δVqq. Given that naively the characteristic energy of a process in diboson production is √ŝ ∼ TeV, one expects that the subleading amplitudes, which grow with √ŝ /m W instead ofŝ/m 2 W , can set bounds that are worse by a factor √ŝ /m W ∼ 10 (as long as the BSM squared amplitudes dominate the cross section, which, as we will see, is the case in our analysis). These subleading amplitudes involve a longitudinal and a transverse vector boson in the final states. For pp → W W , they are given by while for pp → W Z, one has up to subleading terms suppressed by ∼ 1/ √ŝ . One can check that the combination of coefficients entering in the subleading amplitudes cannot be obtained as a linear combination of the directions appearing in the leadingŝ/m 2 W amplitudes. Hence, one naively expects to find some directions in the EFT space that are O(10) times less constrained than the five directions given by the amplitudes leading at high energy. We confirm this naive estimate later in section 3.2 where we study the correlations among the different constraints in the Higgs basis.
To conclude this section, it should be noted that the previous counting is different for universal theories. As discussed in Appendix B, the high-energy diboson amplitudes depend only on five independent parameters and no flat direction is expected in the global fit to diboson data. Anticipating the results that will be presented in the rest of the paper, one should be aware nonetheless that LHC diboson data will not be competitive to LEP-1 to constrain the Zqq couplings in universal theories, at least by using only the current leptonic experimental distributions.
3 Results with current LHC data

Data used and statistical analysis
To get the bounds on the different BSM parameters of Eqs.
(2)-(3), we have used all the leptonic channels of the pp → W W, W Z channels reported by CMS and ATLAS, see Table 1.
We indicate in each case the differential distribution used to perform the combined fit. We limited the analysis to purely leptonic decays due to their high sensitivity and the ease with which one can reproduce the experimental analyses. See Ref.
[43] for a summary of the ATLAS and CMS constraints. There are nonetheless other channels that would be interesting to add, e.g. two quarks and two leptons in the final state [44], since they can set even tighter constrains than the purely leptonic channels. 6 To perform the fit, we calculate the BSM cross sections at tree level with MadGraph5 [45], while using FeynRules 2.0 [36] to generate the .ufo file for the BSMC model [35]. This procedure gives the cross section in terms of the seven BSM parameters δg Zu,d L,R and δg 1z , δκ γ , λ γ . We perform a simulation to get the cross section for each bin for every differential distribution shown in Table 1, and then perform the cuts as described by the experimental collaborations in each case. 7 To get the BSM cross section, we have generated for each bin several simulations corresponding to different values of the BSM coefficients and then we have fitted them to 6 See Ref. [23] where projections for the semi-leptonic channels at HL-LHC are studied in detail, and new experimental observables are proposed. 7 In some cases, like W W → ν ν , the cuts performed by the experiments for some sub-chanels are performed using a Boosted Decision Tree and not just a cut and count approach. In this case we only generate the subchannel for which we can easily reproduce the cuts, i.e. W W → ν e eν µ µ and then fit to the total combination assuming that it does not depend on the lepton flavour. a general quadratic polynomial of the seven BSM coefficients δg Zu,d L,R and δg 1z , δκ γ , λ γ which we schematically call δ i . In other words, we write 8 where the indices i, j go from i, j = 1, ..., 7, where σ SM corresponds to the SM contribution, and a i and b ij are numerical coefficients that characterize the BSM contribution which we determine by varying the BSM parameters δ i , δ j in the MadGraph5 simulation. We then built the ratio δµ defined as Currently, the fully leptonic W W and W Z cross sections have been computed at NNLO in QCD taking into account both on-shell and off-shell contributions [46][47][48] and at NLO in EW but only on-shell [49,50]. If, in the fiducial phase space considered, the effects of taking into account the NLO corrections can be encapsulated by an overall k-factor, 9 the higher order corrections will mostly cancel in the ratios δµ, and that is why we use them to perform the global fit which, as mentioned, is done at LO. This might not hold for transverse polarizations as a result of the non-interfering effects pointed out in Ref. [9], and an analysis can be found in Ref. [10]. Finally, we build a χ 2 function where the first sum runs through all the channels under study, and the second sum runs over each bin for the chosen differential distribution, σ measured is the measured cross section including signal and background,σ bkg SM andσ signal SM correspond to the simulated cross sections for the signal and background done by the experimental collaborations, ∆ syst is the theoretical uncertainty given by the experimental collaborations on the predicted SM cross sections,σ bkg SM andσ signal SM , and finally ∆ stat is the statistical error. When needed, we multiply and divide Eq. (13) by the integrated luminosity squared and compute the χ 2 function using the number of events shown in the figures referred to in Table 1. 8 When simulating the BSM cross sections, we modify the four Zqq couplings δg Zu L , δg Zd L , δg Zu R , δg Zd R , for all the quark generations at the same time, as one would do in the FU case, see Eq. (35). Nonetheless, due to the proton's PDF, the contribution of the light quarks u, d is more than a factor ten greater than the one of c, s, so one can safely assume that the modifications of Zqq for second and third generation give negligible contributions to diboson production. We expect that the results we get for the diboson fit on the Zqq couplings for the FU case also apply for the Zqq couplings for the first two generations of the MFV case, since in the MFV case [δg Zu,d L,R ] 11 [δg Zu,d L,R ] 22 . 9 This would not be the case if the LO amplitude is highly suppressed. This is actually what is happening for the WZ production channel as emphasized in Ref. [12]: in the central region of the detector the ±0 and ±∓ LO amplitudes exactly vanish. We thank G. Panico for pointing this out to us.
From the correlation matrices, central values and errors given in Ref. [27] 10 , we build a χ 2 function for the LEP-1 measurements at the Z-pole. To perform the global fits to get the aTGC bounds, we combine the two χ 2 for diboson at the LHC and LEP-1 as

Correlations among the Higgs basis parameters
When performing a χ 2 fit, in the Gaussian limit, one can easily find the correlation between two parameters by looking at the entries of the correlation matrix. In our case, given that the χ 2 function is not Gaussian due to the non-negligible size of the d = 6 BSM quadratic amplitudes, we cannot easily extract a correlation matrix. Therefore, to get a sense of the correlations among the different BSM coefficients, we perform a global fit and look at the two dimensional plots for each pair of coefficients profiling over all others. We show all these correlations in Appendix A.
As an example of the correlations among the different parameters, in the center of Fig. 2, we show the projection of the χ 2 function onto the two dimensional plane δκ γ , δg Zu R . The least constrained direction in this plot follows the slope given by the combination appearing in the amplitude M(RR; 00) of Eq. (6). The high energy flat direction is about ten times less constrained than orthogonal direction, in agreement with the naive estimate made in section 2.1.
The large correlation shown in the center of Fig. 2 makes δg Zu R and δκ γ very sensitive to each other. For reference, we show in horizontal blue dashed lines the allowed 95% CL bounds set by LEP-1 on δg Zu R and in vertical the 95% CL bounds set by LEP-2 on δκ γ . From this plot one can intuitively see that if δg Zu R is not set to zero but can vary within the range allowed by LEP-1, one may modify the bounds on δκ γ in a non-negligible way. This indicates that the bounds on the aTGCs should include the δVqq deformations if a FU or a MFV scenario is assumed. Also, one can see that the assumptions on δκ γ will have a large impact on the sensitivity of diboson production to δg Zu R . We see that the sensitivity of diboson to the different parameters is ultimately limited by the correlations, making a global combination crucial.
Fortunately, in a broad class of models, the parameter δκ γ is expected to be generated only via loops, and, parametrically smaller than the other parameters, it can be neglected when setting constraints. The same holds true for λ γ which is also typically loop suppressed. This is because both δκ γ and λ γ modify the magnetic moment and electric quadrupole moment of the W which are only generated at one loop in minimally coupled theories [33,51]. Because of the large correlations, setting them to zero can greatly increase the accuracy of the fit to the various δVqq. Leptonic pp→WW,WZ (7,8,13TeV) pp→WW/WZ, 7 param. fit (aTGC profiled) pp→WW/WZ, 4 param. fit (aTGC=0) Figure 2: 68% (dark shaded) and 95% (light shaded) CL regions using the LHC diboson data reported in Table 1. Center: fit to δκ γ and δg Zu R profiling over all other five parameters. The line shows the expected flat direction in theŝ → ∞ limit that can be deduced from Eq. (6). Left (Right): in yellow the fit to aTGC (vertex corrections) marginalising over all other parameters, and in red (pink) the fit when the four δVqq (three aTGC) are set to zero.

δVqq: LHC bounds vs LEP-1 constraints
In Fig. 3, we show the allowed 95% CL regions for the BSM coefficients δg Zu L , δg Zu R , δg Zd L , δg Zd R defined in Eqs. (2) and (3), assuming i) that the aTGC are not negligible (yellow), ii) that λ γ = δκ γ = 0 (blue) and iii) that λ γ = δκ γ = δg 1z = 0 (pink). In gray we show the bounds extracted from the LEP-1 fit of Ref. [27], assuming that the EFT obeys either a MFV (light gray) or a FU (dark gray) flavour structure. To avoid confusion, we remind that when extracting the diboson bounds, we do not differentiate the cases of MFV and FU since diboson production is mostly insensitive to possible differences between the light generations and the third generation that could appear in the MFV case; the only difference is a matter of interpretation, i.e. if one assumes FU the diboson bounds on the Zqq anomalous couplings apply to all the three quark generations, while if one assumes MFV they only apply to u, d, c and s quarks.
We find that even for the most general case that includes all the seven BSM parameters (yellow), the diboson bounds for the down-type couplings are already competitive with those from LEP-1 one under the MFV scheme. The LHC bound on δg Zd R is better than the LEP-1 under the MFV hypothesis and it remains competitive under the FU assumption. On the contrary, for the up type quarks, we find that the LHC bounds are still significantly worse than those from LEP-1, even under the MFV assumption.
Assuming that λ γ = δκ γ = 0 (blue), we find a big improvement on the diboson fit with respect to the seven parameter fit (yellow). The most striking difference being that for the up-type quark couplings, δg Zu L and δg Zu R , the diboson bounds become of the same order of magnitude as those from LEP-1; from these two couplings, it is δg Zu R that benefits the most from setting λ γ = δκ γ = 0. For the down type couplings, we also find an improvement of Leptonic pp→WW,WZ (7,8,13TeV) and LEP-I pp→WW/WZ, (4 param. fit, δg 1 z =δκγ=λγ=0) pp→WW/WZ, (5 param. fit, δκγ=λγ=0) pp→WW/WZ, (7 param. fit)

LEP-I (MFV)
LEP-I (FU) Figure 3: 95% CL regions for the anomalous couplings between the light quarks and the electroweak bosons. In light (dark) gray, the LEP-1 constraints assuming MFV (FU). In yellow, the diboson bounds after profiling over the remaining five parameters. In blue (pink) the same but setting δκ γ = λ γ = 0 (δκ γ = λ γ = δg 1z = 0). about a factor two when setting the two aTGC to zero. With these improvements, the current LHC diboson data set constraints on δg Zd R that are of same order as those derived from LEP-1 data in a FU setup. For MFV scenarios, the LHC bounds significantly outperform the LEP-1 ones.
In pink, we report the constraints for scenarios in which all the three aTGC are negligible compared to δVqq. Actually, letting δg 1z float or not does not significantly change the conclusion: we see that the left handed couplings get a significant improvement with respect to the blue region, while the right handed ones are almost insensitive to this extra assumption on δg 1z . This, again, can be understood from the "correlation matrix" of Fig. 12 that shows that δg 1z is mostly correlated with the left handed couplings.
Note that the correlations among the left and right couplings from LEP-1 measurement are not aligned with the correlation appearing in pp → W V data, which gives some synergetic value to the combination of the two sets of data. This can be seen for example in Table 3 that gives the individual constraints from diboson and LEP-1 and their combination when δκ γ = λ γ = 0. Notice also that while the LEP-1 data for down quarks has a two sigma excess (driven by the Zbb asymmetry) in an analysis in a FU context, the LHC diboson data presents a two sigma excess as well, but in the opposite direction. So the combination alleviates the tension with the SM.
One should remember that the bounds from pp → W V in Fig. 3 only constrain BSM theories where the new particles are above few TeV (see the discussion on the validity of EFT analysis in section 5), while those from LEP-1 apply to theories where the new particles can be as light as O(100) GeV.
To conclude this section, we note that in the fits of the pp → W V data, the quadratic amplitudes appear to be non-negligible, modifying the constraints by a factor ∼ 1.5 − 2 when δκ γ is neglected, and by a larger factor when δκ γ is taken into account, as a result of the correlations identified earlier.
We comment on what it means for the EFT interpretation and possible BSM models in section 5. Figure 4 presents the 95% CL regions for the three aTGC parametrized by δg 1z , δκ γ , λ γ . In red, we show a fit to the three aTGC setting δg Zu L = δg Zu R = δg Zd L = δg Zd R = 0 and profiling over the one aTGC not appearing in the plot. In this case we only use the LHC data from Table 1. In dashed green and dotted blue, we make a fit to the seven BSM parameters, the three aTGC δg 1z , δκ γ , λ γ and the four δg Zu,d L,R , and profile over those not appearing in the plot; in this case we use χ 2 = χ 2 diboson + χ 2 LEP-1 , assuming FU (dashed green) and MFV (dotted blue).

LHC bounds on aTGC and interplay with δVqq
From Fig. 4, we see that the effect of not neglecting the δVqq is the largest in the (δκ γ , δg 1z ) plane, where the constrained area in parameter space varies around 50% from one assumption to the other. This points to a large correlation between δκ γ and δg 1z on the δVqq parameters, which is to be expected since they appear in the same high energy amplitudes as seen in Eqs. (6) and (7). The determination of λ γ is insensitive to the different assumptions, as expected from the fact that it is the only parameter appearing in the amplitudes that grow withŝ and have final polarizations ±±.
Given that in many BSM models δκ γ and λ γ are assumed to be loop induced and therefore parametrically smaller than δg 1z , we also study the effect of profiling over δVqq when δg 1z is the only aTGC modifying the diboson production. Since the global fit is non-Gaussian, this particular case with δκ γ = λ z = 0 cannot be obtained simply from the general case. In Fig. 5, we show in solid black the one parameter exclusive fit to δg 1z , setting all the other parameters to zero. In dashed green and dotted blue, we allow δg Zu L , δg Zu R , δg Zd L , δg Zd R to be different than zero and perform a global fit.
Similarly to Fig. 4, Fig. 5 tells us that profiling δVqq in the fit changes the current constraints on δg 1z by a factor of about 25%. Also, we find that once the δVqq are introduced, the FU and MFV assumptions which modify χ 2 LEP-1 yield qualitatively similar size effects to δg 1z but still with at least 10% differences between the two.
In this section, we have presented an analysis using all the current leptonic diboson data to set constraints to both aTGCs and δVqq vertices, and study the correlations under different flavour schemes. The LEP-1 constraints for δVqq in Fig. 3 could lead to the conclusion that for a MFV setup larger deviations could be obtained. However, this is not the case because the cross correlations among LEP-1 and LHC diboson data make the global fit more constraining than both sets of data alone. The largest correlation in the fit appears to be between δκ γ and δg Zu R as established in section 3.2.

Data used and assumptions for HL-LHC
To estimate the bounds at HL-LHC, as a first step and for simplicity, we simulated the channels pp → W W → ν ν and pp → W Z → ν . We build a χ 2 function with the same form as in Eq. (13) and we inject the SM signal, i.e., we assume that the measured number of events will be the same as in the SM prediction, so that σ measured =σ bkg SM +σ signal SM defined after Eq. (13). Therefore the χ 2 can be written as: where we define δ syst = b + (L ∆ syst ) 2 /s, with ∆ syst being the absolute systematics error in the cross section, L being the integrated luminosity, and δ stat = 1/ √ s. s and b stand for the number of simulated SM signal and background events, and δµ( δ ) is defined in Eq. (12). As usual, events with misidentified particles, such as misidentified leptons in processes with W +jets or top production (see e.g. section 5 of Ref. [39]), are included within the background.
There has been no extensive study of the systematic uncertainties and the expected background for pp → W V , especially in the high energy bins. A 5% of systematic uncertainties is claimed to be possible in Ref. [12] in the fully leptonic W Z channel within the fiducial region used for their analysis, and is used as a benchmark in Ref. [23] for the semileptonic W V and W h channels. In Ref. [4], instead, it is claimed that this accuracy can only be reached by measuring ratios of cross sections. We take a pragmatic approach and consider two scenarios for the uncertainties at HL-LHC: a pessimistic one where δ syst = 30% is assumed for all the bins, which corresponds to an extrapolation of the uncertainty in the overflow bins of the experimental analysis, and a more aggressive scenario where one assumes δ syst = 5% for all bins.
For the W W channel, we consider the m distribution and for the W Z channel, we consider the p Z T distribution. In both cases we have chosen the binning in such a way that the overflow bin contains ten events. As a small cross check, we compared our estimated bounds on the aTGC at HL-LHC with 3 ab −1 , shown in red in Fig. 7, with those in Fig. 3 of Ref. [52]. There the channels W γ → νγ and W Z → ν were considered and bounds on the aTGC were derived for a run at 14 TeV with a total accumulated luminosity of up to 1 ab −1 . Our bounds, assuming δ syst = 5% in the leptonic W W and W Z channels, turn out to be more conservative than those in Ref. [52] but overall of the same order. So, our simple assumptions are in line with the existing literature and should give a reliable and conservative estimate of the HL-LHC reach. Note that there are several ways to improve the diboson analysis: i) the semileptonic channels can be considered on top of the purely leptonic ones, ii) more refined observables like those presented in Refs. [9,11,12] can be studied. Therefore even with δ syst = 5%, our estimates on the diboson reach at HL-LHC are probably on the conservative side. To compare the traditional experimental analysis with new proposals, in section 5.4 we  Figure 6: Estimated 95% CL bounds at HL-LHC on the anomalous couplings between the light quarks and the electroweak bosons. In yellow, diboson bounds after profiling over the remaining five parameters. In blue (pink), same but setting also δκ γ = λ γ = 0 (δκ γ = λ γ = δg 1z = 0). Solid and dashed stand for an assumed δ syst = 5% and δ syst = 30% respectively. Light (dark) gray regions correspond to the LEP-1 bounds assuming MFV (FU).
compare the HL-LHC reach of leptonic W Z estimated in Ref. [12] with our combination of the leptonic W W and W Z using the m and p Z T differential distributions. Figure 6 shows the allowed 95% CL regions for δg Zu L , δg Zu R , δg Zd L , δg Zd R in the three different scenarios: i) the three aTGC, λ γ , δκ γ , δg 1z , are kept as floating parameters in the fit (yellow), ii) λ γ and δκ γ are set to zero (blue), and iii) the three aTGC are set to zero (pink). A total accumulated luminosity of 3 ab −1 is assumed. In order to appreciate the improvement compared to LEP, the gray regions report the bounds extracted from the LEP-1 data under the MFV (light gray) and FU (dark gray) assumptions. Clearly, for low enough systematics, HL-LHC will surpass the LEP-1 bounds for any new physics scenario with a built in MFV structure that does not generate anomalously large aTGC, i.e. scenarios for which δκ γ = λ γ = 0 (blue) is a good approximation. Under the FU assumption, the HL-LHC bounds on δg Zu R and δg Zd R vastly surpass the LEP-1 bounds whenever δκ γ = λ γ = 0, while the bounds on δg Zu L and δg Zd L are only slightly better. In any case, it should be noted that the blue and pink bounds improve by one order of magnitude at HL-LHC compared to the current bounds. As long as the systematics remain low enough, the seven parameter FU fit also improves by about a factor three the bounds for all the δVqq with respect to the current bounds shown in Fig. 3. The seven parameter FU fit equals or surpasses the LEP-1 constraints for δg Zd L , δg Zu L and δg Zd R . On the other hand, with higher systematic uncertainties, δ syst = 30%, the improvement from the seven parameter and five parameter fits with respect to the current constraints will be limited and mostly concern the right handed couplings. Only for δg Zd R , the HL-LHC will show an improvement over LEP-1 in all the cases, both for MFV and FU structures. Figure 7 shows the allowed 95% CL regions for the three aTGC parametrized by δg 1z , δκ γ , λ γ . In red, we show a fit to the three aTGC setting δg Zu L = δg Zu R = δg Zd L = δg Zd R = 0 and profiling over the one aTGC not appearing in the plot. In green, we make a fit to the seven BSM parameters, namely the three aTGC δg 1z , δκ γ , λ γ and the four δg Zu,d L,R and we profile over those not appearing in the plot. We use the HL-LHC projections to build χ 2 diboson while the χ 2 LEP-1 is built from the global fits performed in Ref. [27]. We find that at HL-LHC the differences between assuming MFV or FU for χ 2 LEP-1 are negligible when performing a combined global fit of LEP-1 and LHC data. For this reason in this section we only present results with the FU hypothesis for the LEP-1 fit.

HL-LHC projections on aTGC and interplay with δVqq
At HL-LHC, the aTGC bounds shown in Fig. 7 are qualitatively similar to those of Fig. 4 obtained with the current data. The main difference between the two is that the features found with the current data regarding the impact of δVqq are accentuated at HL-LHC. This is particularly true for δκ λ and δg 1z : the bounds on the δκ γ , δg 1z vary by more than 100% if instead of setting δVqq = 0 they are included in a global fit combining the LEP-1 data in the context of FU or MFV scenarios. On the other hand λ γ will remain mostly unaffected, as anticipated from Eqs. (6) and (7). LEP-1 data and the light quark vertices are profiled over. The bound on δg 1z is rather robust and does not show a strong dependence on the assumed systematic uncertainty, changing by a factor two between when the systematics vary from 0% to 50% (the statistical uncertainty is of course kept). The HL-LHC bound will be of the order of 0.1%, an order of magnitude better than the current existing bound. And further improvement can be anticipated, e.g. by relying on the new analyses proposed in Ref. [12].

Interpretation of the constraints
In this work we have performed a global analysis of the diboson data at the LHC and inferred bounds on aTGCs as well as on anomalous couplings of the quarks to the EW gauge bosons. We found that in some cases these bounds surpass the LEP-1 and LEP-2 bounds. Nonetheless, it is important to stress that this is only so for certain regions of the parameter space. As in any EFT analysis, the constraints on the Wilson coefficients are only valid when the characteristic energy of the processes remains smaller than the masses of the new particles. Furthermore, both for the current LHC data and also for the HL-LHC ones, the quadratic terms of the BSM contributions to the diboson production cross section play a non-negligible role in settings bounds on the Wilson coefficients. In that situation, further restrictions on the parameter space follow to ensure that the interference between the SM amplitude and the dimension-8 operators, formally of the same order as the square of the dimension-6 operator contributions, remains sub-dominant [5-7, 9-12, 15, 17-22]. We comment on these two limitations, in the following, and also see how they appear in a concrete toy model with vector triplets.

Quadratic BSM amplitudes
As already noted and extensively discussed in Refs. [5][6][7][9][10][11][12]15,[17][18][19][20][21][22], when setting bounds to the EFT coefficients, it may happen that these bounds only constrain BSM amplitudes that are larger than the SM one. This makes the quadratic dimension six BSM amplitudes to be non negligible. To get a sense of which BSM theories can be studied only using dimension six operators while neglecting those of dimension eight, it is useful to schematically write the ratio of amplitudes between the EFT and the SM. These estimates have already been discussed in Refs. [9,17,32], here we only give a small review for convenience. Schematically, for the W V channels with longitudinally polarized gauge fields, the ratio of EFT and SM amplitudes is given by where c 6 , c 8 represent the coefficients in front of the d = 6, 8 operators. When the quadratic terms dominate, the following condition has to be fulfilled in order to be able to neglect the dimension-8 operators For simple power counting rules such that c 6 ∼ c 8 ∼ g 2 , with g a charactestic coupling of the new physics degrees of freedom, the EFT validity condition simply requires that the BSM coupling must be larger than the SM one, g 2 g 2 SM , which is nothing else than the condition that also ensures that the quadratic BSM pieces dominate Eq. (15).
In the channels with mixed longitudinal and transverse polarizations, for which the new physics amplitude only grows as √ŝ , the same conclusion applies. The channel with transverse polarizations only is, however, slightly different. In that case, the linear/interference terms at the dimension-6 level is suppressed due to the necessity to go through a helicity flip [32]: And the quadratic pieces can dominate the linear terms for smaller values of g . One would then end up in a region of the parameter space where the EFT analysis would not be valid since the dimension-8 operators cannot be neglected. Recently, new observables have been proposed [9,11] to resurrect the interference and then circumvent this (in)validity issue.

Power countings and BSM interpretations
Assessing the consistency of EFT interpretation requires to assumptions on the scaling of the Wilson coefficients of the higher dimensional operators. We present here three different power counting and selection rules which inspired the particular choices of the BSM parameters kept in the fits presented in sections 3 and 4. They correspond to specific dynamics for the new physics above the weak scale.
• In many renormalizable UV models with heavy resonances or SILH like models [51], λ γ and δκ γ are loop suppressed while δg 1z is generated at tree level, ending up in the following power counting for the aTGCs: For this broad class of scenarios, δκ γ and λ γ can be safely neglected since they are parametrically suppressed with respect δg 1z . Even if the fermion have no direct couplings to the UV sector, some δV qq deviations will be generated in partial compositeness scenarios when the light quarks mix with fermionic resonances Ψ belonging to some strong sector via q m qΨ. This induces a coupling of the light quarks to the strong sector that involves in particular the longitudinal polarisations degrees of freedom of the EW gauge bosons. This coupling is proportional to the strong coupling g and the mixing parameter q : • In the limit q 1 of the partial compositeness scenario, the quarks can be considered purely elementary. The contribution (19) becomes subdominant and one is left with the contribution originating from the composite interactions of the gauge sector that amounts to: which is comparable in size to the deviation expected for δg 1z , while δκ γ and λ γ can be safely neglected. However, in this elementary limit, the dynamics naturally flows to the case of a universal theories for which there exist correlations among the four δVqq, see Eq. (36), and the constraints on the parameter space of such models should properly take into account these correlations.
• Finally, there are models where the three aTGCs can be large and should be kept in the global fit of the diboson data. This is the case for instance of the Remedios + ISO(4) scenario proposed in Ref. [53], where the transverse gauge bosons belong to the composite sector too. The UV dynamics is such that λ γ , δκ γ and δg 1z are generated at tree level and obey the the following power counting with g g SM .

Energy limitation
An EFT has an intrinsic cutoff scale and for the analysis to be valid it should not use any event with a characteristic scale above this cutoff. Given that the center of mass energy of the interacting partons is not known at the LHC, it may be impossible to know the center of mass energy of a given process if the energy and momentum of the final states are not completely reconstructed. This is the case of the leptonic processes for pp → W V with one or two neutrinos in the final states. We set 3 TeV as the energy for which the EFT stops to be valid following the analysis from Ref. [6] and checking that we did not get any events in the Madgraph5 simulation above 3 TeV. One could extend the EFT reach below the 3 TeV mark without changing the experimental analysis by following the procedure explained in Ref. [15,17,54,55]. This procedure is based on considering only the events with a characteristic energy below a pre-determined cutoff scale E cut . The constraints obtained this way are, although not optimal since one is throwing away the events above the cutoff, totally consistent with the EFT expansion.
To illustrate the effect of the energy limitation to stay within the validity region of an EFT analysis, one can consider the projections of the δVqq bounds onto the parameter space (g , m ) on models whose dynamics follows the power counting discussed above.
• For SILH-like models with elementary quarks, the scaling (20) naively leads to a simple 68%CL lower bound on m independent of g : m > (500, 900, 1300) GeV at LEP-1, LHC, HL-LHC respectively (22) (we used the constraints on δg Zd R of Tables 3 and 5, under the MFV assumption and setting the aTGC to zero). However, the bounds in Eq. (22) for the LHC and HL-LHC are not reliable since they fall outside the regime of validity of the EFT. Hence, only the LEP-1 bound can be trusted for these types of models. Futhermore, as commented above, in this elementary quark SILH scenario, there exist some correlations among the four δVqq couplings and a more meaningful bound on (m , g ) should take into account these correlations. For instance, the 95%CL LEP-1 bound can be obtained from the universal theory fit, see Eq. 37, leading to In deriving this constraint, we have used δg Zd R ∼ g 2Ŝ /(3g 2 − 3g 2 ) and assumed the scalingŜ ∼ m 2 W /m 2 , whereŜ is one of the oblique parameters relevant for universal theories.
• For the SILH-like models with composite quarks, the situation is different and there the diboson channels at the LHC can be used to set reliable constraints stronger than the ones derived at LEP. Indeed the scaling from Eq.
And this time, the validity constraint of m > 3 TeV implies that at the LHC and HL-LHC, diboson data can reliably constrain theories with g > 2.1 and 1.2 respectively, i.e., with a characteristic coupling slightly larger than the electroweak one.

A model with triplets: diboson reach vs other searches
In this section we put the previous results in a global perspective, assessing the usefulness of diboson observables in a simple UV toy model where other types of searches are also constraining the parameter space. Our motivation stems from the fact that from an EFT point of view, non-universal corrections to the light quark vertices come from operators of the type (f γ µ f )(H † ← → D µ H), and in general grounds one expects to also generate the operators (f γ µ f )(f γ µ f ) and (H † D µ H)(H † D µ H), which affect dijet processes and Higgs physics respectively. Considering a particular model allows one to compare these different searches and appreciate their complementarity. 11 We focus our attention to the general vector triplet models presented in Refs. [15,56,57], which appear in various BSM scenarios, and can produce sizable and non-universal deviations to δVqq for the light quarks. We will see how the different searches are sensitive in the different limits of the parameter space.
For generality, we give the expressions for a model with custodial symmetry consisting on two vectorial resonances, L µ and R µ , transforming respectively as (1, 3, 1) and (1, 1, 3) under SU (3) C ⊗ SU (2) L ⊗ SU (2) R . At leading order, these resonances couple to the SM currents as follows: where the SM currents are given by The simplified UV model is fully characterized by the 11 arbitrary parameters (γ H , . The couplings to each fermion also carry flavour indices. In the following, we will assume that they follow the MFV flavor scheme, with the two lighter generations having roughly the same γ f and δ f couplings, and the third generation being different. When both resonances have a mass m m W , they can be integrated out to generate higher-dimensional interactions among the SM particles. At order 1/m 2 , see Ref. [15], this yields (27) where the operators are defined as At tree-level, the matching between the UV model and its EFT description leads to the following expression of the Wilson coefficients appearing in Eq. (28) In Eq.  Table 97 of Ref. [26]. Specifically, we have 12 ,

3.
γ H γ Q Figure 9: 95% CL exclusion regions in the (γ H , γ Q ) plane for various values of the resonance masses. In blue, the projected constraints from diboson data at HL-LHC, using our projections (dark shade) or the refined analysis strategy of Ref. [12]. The other constraints come from a recast of the studies in Refs. [21,[59][60][61][62] and are commented in detail in the text.
The operators O f f , O (3) f f and O H do not contribute to diboson production but they modify dijet and Higgs production which can then be used to set constraints on the parameters γ f , δ f , γ H , δ H of the simplified UV model.

General models with only L µ
In order to compare with experimental bounds and previous works, we will only consider the scenario where one has only the L µ resonance. We further assume for simplicity that γ V γ H , γ Q . This minimal setup interpolates between the strongly and weakly coupled limits in the Higgs and fermionic sectors. Figure 9 shows, for fixed m , the constraints on (γ H , γ Q ) from various searches. The blue regions correspond to constraints imposed by the future diboson measurements at HL-LHC, using our projections (dark shade) or the refined the analysis strategy proposed in Ref. [12] (light shade). For comparison, the LEP-1 bound taken from Ref. [27] under the MFV flavour scheme is indicated by the dashed light blue line. For this scenario, HL-LHC hardly competes with LEP-1. The systematics uncertainty in diboson measurements would have to go significantly below 10% to overcome the LEP-1 constraints.
For light resonance mass, m = 5 TeV (left plot), the parameter space is severely constrained by direct resonance searches in the leptonic channel (yellow), hadronic channel (red) or diboson channel (dark red). The bounds have been obtained by recasting the projections of Refs. [59,60]). To derive the bound from Drell-Yan searches, we set γ = γ Q . The bright red dotted line delineates the boundary between the regions in which L µ has a width smaller or larger than 20% its mass; this separates the regions where the direct searches may stop being sensitive to these resonances (at large γ Q and γ H ). Finally, the sensitivity at HL-LHC in the Higgs coupling measurements also cuts off the region with γ H > 9, corresponding to ξ = v 2 g 2 /m 2 > 0.08 [59,61,62].
At higher resonance masses, m = 7 TeV (center plot) and 10 TeV (right plot), the direct resonance searches loose steam and the diboson channels become more relevant in a larger portion of the parameter space. Already for m = 7 TeV, the resonant dijet bound falls in the region where Γ/m > 20%, questioning its validity. The Higgs and resonant diboson constraints are too weak to set any constrain at these masses. For m = 10 TeV, the constraints from resonant Drell-Yan searches fade away too.
Non-resonant dijet observables also impose severe constraints on the viable parameter space of our simplified model, the dijet EFT lines in Fig. 9. We used the results of Ref. [21] that puts bound on coefficient Z of the dimension-6 operator involving two gluon field strenghts Using the equations of motion, this operator can be rewritten in terms four-fermion operators.
The results presented in this section can be translated to the benchmark models A and B suggested in Ref. [57]. The model A corresponds to a gauge bosons from an extended gauge symmetry and it features γ H γ f ∼ g 4 /g 2 and γ H /γ f ∼ 1. Figure 9 shows that this scenario is better probed via direct searches. The model B corresponds to a resonances from a composite sector, and it features γ H γ f ∼ 1 and γ H /γ f ∼ g 2 /g 2 , which projects in parameter space onto a line parallel to the indirect diboson constraints in Fig. 9. The indirect probes can bring information complementary to the direct constraints.
In conclusion, the diboson channels give interesting constraints in regions where γ Q is small while γ H is large, which can be mapped to composite models with heavy resonances strongly coupled to the Higgs boson but weakly coupled to the light quarks. In this section, we only studied the case with a left handed resonance L µ in order to compare with the direct experimental searches and with previous phenomenological works. We find that, with an uncertainty δ syst = 10%, the pp → W W, W Z leptonic channels at HL-LHC will access regions of the parameter space that remain blind to other searches. Nonetheless, for this simplified scenario, LEP-1 is still slightly better than our HL-LHC projections. It would be interesting to study the case where more than one resonance is present, and therefore the various δVqq are less correlated. We expect, as shown in sections 3.3 and 4.2, that in these cases diboson production will be significantly better than LEP-1 while being complementary to direct searches.

Summary and outlook
The high energies accessible at the LHC open the possibility not only to directly produce new states, but also to enhance the sensitivity to new physics out of direct reach with effects that are encoded in higher dimensional operators involving the SM degrees of freedom. We offered a detailed analysis of diboson processes at LHC, which provides an interesting probe of some In yellow we use LHC diboson data and perform a global fit including the aTGCs. In blue, we only profile over δg 1z and the vertex corrections. The thicker boxes combine LHC and LEP-1 data.
of these operators, in particular those that give rise to effects growing with the characteristic energy scale of the underlying hard process.
Due to the expected increased sensitivity in the analyses, we reiterated that the interpretation of the diboson measurements in terms of anomalous triple gauge couplings has to be reconsidered. In particular, the effects of anomalous couplings among the light quarks and the electroweak bosons can no longer be neglected a priori. On one hand, the current LHC diboson data already set stronger constraints than LEP-1 on the anomalous couplings δVqq for the down quark, at least under the hypothesis of MFV. On the other hand, both in the MFV and FU hypotheses, the aTGC fit is found to be only marginally stable under profiling over the δVqq vertex corrections even when the LEP-1 constraints are imposed.
We did a simple estimate for the HL-LHC reach and found that the constraints will improve by a factor two to three. The different flavour assumptions on the vertices will have a seizable impact on the aTGC constraints. Quite remarkably, the precision on light quark couplings at HL-LHC will significantly surpass the LEP-1 constraints for both MFV and FU assumptions. And, as shown in Fig. 8, the HL-LHC may be able to set bounds on δg 1z of the order of 0.1% in both FU and MVF scenarios. On the contrary, we checked that for universal theories which, as shown in Appendix B, depend only on three aTGC and two δVqq, the HL-LHC bounds are still far from reaching the LEP-1 precision. Precision ×10 2 95%CL, leptonic pp→WW,WZ at 13TeV with 3ab -1 and LEP Figure 11: Current (left) and future (right) 95% CL constraints on δg 1z . In gray, LEP-2 results. In blue, constraint from LHC diboson data alone. In green, fit to LHC diboson data including the anomalous Vqq vertices and profiling over them with the LEP-1 MFV constraints. In the projection of the HL-LHC bounds, a 10% systematic uncertainty in the channel pp → W W, W Z is assumed.
The left plot of Fig. 10 shows that the current leptonic diboson data can already set bounds setting bounds on δVqq that are competitive with LEP-1, and they can also improve the bound on δg Zd R . The right plot shows that, by the end of HL-LHC, leptonic pp → W V can be very competitive with the LEP-1 bounds or greatly surpass them if one assumes that δκ γ = λ γ = 0. Focusing on the aTGCs, Fig. 11 tells that, even when δκ γ and λ γ are neglected, δg 1z is quite sensitive to the δVqq anomalous couplings: both with the current data and at HL-LHC, the δg 1z bounds varying by about 30% when δVqq are switch on and off in the global fit.
We studied the interplay between the operators probed in diboson and the ones probed in other searches, as dijets or Higgs physics. This interplay can be intuitively understood by remembering that the operators affecting diboson take the form (f γ µ f )(H † D µ H), and one can expect that generically might be accompanied by operators like (f γ µ f ) 2 and (H † D µ H) 2 as well. As a concrete example, we presented a model in which all those deviations are indeed induced, showing that measurements in diboson offer a complementarity exploration of the parameter space. It would be interesting to see how the direct and indirect bounds change for models with more resonances; we expect that diboson will fare better compared to the other searches when considering less simplified scenarios.
There are several interesting future directions. Focusing on the current experimental searches, it would be interesting to study the semileptonic channels, which might benefit from fat jet techniques [44], perhaps allowing to reach higher invariant masses than the leptonic ones. Regarding new searches, one could follow the steps advocated in Refs. [9,11,12,23] and find new ways to increase the sensitivity to certain BSM physics allowing for more general interpretations of the bounds while also lowering the mass scale which one can probe. Regarding the results presented in this paper, we would like to encourage the experimental collaborations to use the current diboson searches to set bounds on the anomalous couplings between the light quarks and the Z boson. It would also be very interesting to see how the degeneracy between aTGC and δVqq can be resolved by considering the production of a Z in association with two jets by vector boson scattering. In any case, we want to stress that, beyond the case of universal theories, there exist flavour scenarios for which robust bounds on the aTGCs can only follow from a global fit that include the effect of the δVqq anomalous couplings.

A Correlations in the Higgs basis
In Figs. 12 and 13 we show the correlations between all the seven parameters relevant for diboson production at the LHC. Since the χ 2 function is not gaussian, these correlations are not simply related to a covariance matrix. Instead, the 95% CL regions for each pair of parameters with all others profiled are reported. See section 3.2 for comments.
Correlations for the current LHC data Figure 12: One and two dimensional 95% CL constraints for the seven parameters entering in diboson processes, using only the current LHC data in Table 1. In yellow, all parameters are profiled. In blue, we profile over all parameters but setting δκ γ = λ γ = 0. In pink, we do an exclusive fit setting to zero all parameters that do not appear in the plot labels.
Correlations expected at HL-LHC Figure 13: One and two dimensional 95% CL constraints for the seven parameters entering in diboson processes, using the pp → W + W − and pp → W Z projections for 13 TeV with 3 ab −1 of integrated luminosity and assuming a 10% systematic uncertainty. In yellow, all parameters are profiled. In blue, we profile over all parameters but setting δκ γ = λ γ = 0. In pink, we do an exclusive fit setting to zero all parameters not appearing in the plot labels.

B Summary of LEP-1 bounds
In this appendix, we present the LEP-1 constrains obtained by profiling the χ 2 function obtained by Ref. [27].

Minimal Flavour Violation
In MVF scenarios, the vertex corrections have the following form: where i, j = 1, 2, 3 stand for the family index. We are only interested in the constraints on the light quarks u, d that control the diboson production. Using the results in Ref. [27] and after profiling over all other parameters related to the electron and neutrino couplings, we arrive at In this flavour scenario, the vertex corrections are mostly sensitive to the A coefficient in Eq. (32), while the contribution from B, being suppressed by m u,d /m t,b , is negligible. The same bounds will also apply to the c and s quarks since the B contribution remains negligible for the second family.

Flavour Universality
In FU scenarios, all the vertex corrections have the same value irrespective of their family index, i.e.
In this case the bounds for the light quarks and heavy quarks coming from LEP-1 are the same. Using the results of Ref. [27] and after profiling over all other parameters, the LEP-1 bounds on the vertex corrections are found to be In this case, diboson production will set bounds on all of Zqq from just measuring the vertices for u and d. It should be noted that, while the bounds on the Zūu couplings are rather similar in the two MFV and FU cases, the bounds on the Zdd couplings are about 4 times more stringent in the FU case compared to the MFV case. This is a result of the fact that the b quark can be efficiently tagged and better discriminated than the light quarks. On the other hand, for the case of MFV, the Zbb vertex correction gives a good constraint to the parameters A + B in Eq. (32), while δg Zd L,R 11 is only sensitive to A and has a much lower precision from the Z-pole observables.

Universal theories
For universal theories where new physics coupled to the SM degrees of freedom via the SM currents only, the vertex corrections obey the relations 13 and only two Zqq couplings are independent. We can choose them to be δg Zu L and δg Zd L . From the χ 2 function corresponding to FU theories, one can derive the bounds on these two independent couplings In this scenario, the current diboson data do not set competitive bounds on the Zqq couplings. For completeness we show in the following the connection between δg Zu,d L,R , δg 1z and the oblique parameters when considering universal theories. For δg Zu,d L,R one finds: which actually holds for any SM fermion. The two relations 36 are trivially satisfied. In addition, δg 1z can be written as: where the oblique parameters are obtained from the coefficients of the d = 6 operators in the SILH basis:

C Cross checks of the aTGC bounds
As a cross check of our methodology and our assumptions, we compared the results of our fit with the ones presented by the experimental collaborations. For the pp → W W channel at 8 TeV, Fig. 14 shows the comparison between the fit of Ref. [38] by the ATLAS collaboration and the results we obtained recasting the publicly available data. There is a good agreement.
To compare with ATLAS results, we performed a change of basis and set bounds on the coefficients c W W W , c W and c B corresponding to the following three operators which appear for instance in the HISZ basis, see Refs. [25,33]: The Wilson coefficients of the HISZ operators entering the aTGC are related to Higgs' basis coefficients as follows [25]: In the pp → W Z channel at 8 TeV, we could not reach a similar agreement with the ATLAS results reported in Ref. [40], but we do agree with previous phenomenological studies [5,6]. ATLAS pp→WW s =8TeV,20.3fb -1 Figure 14: Comparison between the 95% CL contours obtained by the ATLAS collaboration [38], and the results we obtained recasting their data.
In Fig. 15 we compare the fit on the aTGCs using the LHC diboson data reported in Ref. [38], after profiling over the δV qq couplings, with the results in Ref. [10]. In our work, the aTGC bounds are derived from a global fit to the LHC diboson data and the χ 2 extracted from Ref. [27] for LEP-1. In dashed blue, we show our three parameter fit, which agrees well with the experimental results, as shown in Fig. 14 already. Our results and those from Ref. [10], shown in botted black, are very similar. When the parameters δVqq are profiled using the LEP-1 constraints under the FU assumption, our results (solid blue) show deviations with respect those from Ref. [10] (solid black). The slight differences with the fit from Ref. [10] are due to the following: i) only the last bin of the experimental distribution is used in Ref. [10] while we use all of them, ii) the procedure itself to set the bounds for the aTGC in Ref. [10] is different which could also create some discrepancy with our results. To asses the first point, we show in red our fit after profiling over the quark couplings when only the last bin is used. We find that this has a better agreement with Ref. [10], nonetheless not taking into account the subleading bins spoils our agreement with the ATLAS result.  Figure 15: 95% CL contours for the aTGCs marginalizing over all other parameters. A three parameter fit (dashed or dotted) with δVqq = 0 and a seven parameter fit (solid lines) combined LHC diboson data and LEP-1 data are performed. For the later fit, a FU setup is considered. The blue lines are our results, in black the results obtained in Ref. [10], and in red, our results taking into account only the last bin.

D Comparison of HEP parameter bounds
To compare with previous works studying diboson production, in Fig. 16 we present the HL-LHC bounds for the high energy parameters (HEP) defined in Ref. [12]. These HEP appear in the helicity amplitudes of Eqs. (6), (7). In order to rewrite the Higgs basis in terms of the HEP, we perform a change of basis to in χ 2 function inverting the following relations: q ffrom the HL-LHC projections and comparison with the 95% bounds obtained in Refs. [12,23].
The χ 2 function then becomes a function of the four HEP, and λ γ , and two other orthogonal combinations which we call b 1 , b 2 . These orthogonal combinations appear in the subleading amplitudes shown in Eqs. (9), (10). Figure 16 shows in red and blue the derived χ 2 function for a (3) q assuming δ syst = 5%. 14 The blue and red colours correspond to the case where all the bins of the differential distributions are used (blue) and the one where only the last bin is used (red). Clearly, the actual bound is not entirely dominated by the most energetic bin and all the bins do contribute to setting the bound. We explicitly studied three different cases: i) in dashed, we set λ γ , the three remaining HEP and the orthogonal directions b 1 , b 2 to zero, ii) in solid we set b 1 , b 2 and λ γ to zero but profile over the three remaining HEP, iii) in dotted we profile over all the parameters. We find that, as expected, the four HEP parameters are not very correlated among them, and therefore the solid and dashed lines differ by a small amount. On the other hand, including or not the subleading terms b 1 , b 2 , which appear in the amplitudes shown in Eqs. (9), (10), makes a significant change. In general, we expect the subleading terms to be relatively important when the quadratic dimension-six amplitudes dominate the interference with the SM. On the other hand if the interference with the SM dominates, we expect these pieces to have an extra suppression coming from the SM amplitudes. In vertical orange and green lines we present the HL-LHC prospects of the leptonic pp → W Z and semi-leptonic pp → W V obtained in Refs. [12,23]; we differentiate in their case with solid and dashed lines two different assumptions on the systematic errors. If the new observables proposed by Refs. [12,23] are implemented, they will be able to set stronger bounds on a parameters, i.e. three HEP, b 1 , b 2 and λ γ are the following: On the other hand, Ref. [12] gets ∆a Another way to compare with previous works is to set bounds on the a (3) q , a (1) q plane but considering only universal theories where the oblique parameters W, Y are negligible, see Refs. [12,16]. In the SILH basis using the conventions in Ref. [25], one obtains therefore, for universal theories and neglecting W = −2c 2W and Y = −2c 2B the HEP are: which in terms ofŜ, δg 1z and δκ γ , using the conventions in Ref. [25], can be written as: In Refs. [12,16] for convenience they choose as independent directions δg 1z c 2 W + g 2 g 2 −g 2Ŝ and δκ γ −Ŝ. We show our bounds in this plane in Fig. 17.

E High energy amplitudes in the Warsaw basis
Using the dictionary of Ref. [25] we express the amplitude shown in Eqs. (6)-(7) in the Warsaw basis. The high energy amplitudes for pp → W W are given by:  Figure 17: 95%CL constraints on universal theories with W, Y 1. Left: Comparison of the LEP constraints with the ones extracted from the current LHC diboson data (dotted) and the HL-LHC projections (solid). Right: Constraints of this work compared with the ones in Refs. [12,16]. This fit for universal theories agrees with the 3-parameter fit of Fig. 7 when only the aTGC couplings are considered and the δVqq deviations are set to zero.
while for pp → W Z, these are:

F Summary tables
In this appendix, we report the results of the various fits performed in this paper.

Constraints on δVqq
We first report the bounds on δVqq under various assumptions on the aTGCs.  Table 2: Constraints (×10 3 ) on the δVqq vertex corrections from a seven parameter global fit combining LHC diboson data and LEP-1 measurements. The first column gives the bounds using the LHC diboson data alone. The second and third columns report the LEP-1 bounds derived in Ref. [27] under the MFV and FU assumptions respectively. Finally, the last two columns show the combination of the current LHC and LEP-1 data for the two flavour assumptions.
1σ bounds on δVqq from current LHC data (δκ γ = λ γ = 0)   Table 3: Constraints (×10 3 ) on the δVqq vertex corrections from a five parameter global fit combining LHC diboson data and LEP-1 measurements, setting δκ γ = λ γ = 0. The first column gives the bounds using the LHC diboson data from Table 1 setting δκ γ = λ γ = 0. The first column gives the bounds using the LHC diboson data alone. The second and third columns report the LEP-1 bounds derived in Ref. [27] under the MFV and FU assumptions respectively. Finally, the last two columns show the combination of the current LHC and LEP-1 data for the two flavour assumptions.
1σ bounds on δVqq from current LHC data (δg 1z = δκ γ = λ γ = 0)   Table 4: Constraints (×10 3 ) on the δVqq vertex corrections from a five parameter global fit combining LHC diboson data and LEP-1 measurements, setting δg 1z = δκ γ = λ γ = 0. The first column gives the bounds using the LHC diboson data alone. The second and third columns report the LEP-1 bounds derived in Ref. [27] under the MFV and FU assumptions respectively. Finally, the last two columns show the combination of the current LHC and LEP-1 data for the two flavour assumptions.
1σ bounds on δVqq expected at HL-LHC ×10 3 aTGCs profiled no loop (δκ γ = λ γ = 0) no aTGCs exclusive fit  Table 5: Expected constraints (×10 3 ) at HL-LHC on the δVqq vertex corrections. The constraints are obtained from the projections at HL-LHC for the pp → W + W − → ν ν channel combined with the LEP-1 constraints for a MFV setup. The first column gives the constraints resulting from a seven parameter fit. In the second, the two aTGCs usually generated at the loop level are set to zero. In the third column, all the three aTGCs are set to zero. Finally, the last column reports the constraints obtained from an exclusive fit with only one parameter considered at a time.

Constraints on the aTGCs
We now report the bounds on the aTGCs under various assumptions on δVqq.
1σ bounds on aTGC from current LHC data λ γ 0 ± 6 0 ± 6 0 ± 6  Table 7: Constraints (×10 3 ) on the anomalous triple gauge couplings using the projections at HL-LHC of the pp → W + W − → ν ν channel. The first column corresponds to the traditional diboson analysis that considers only aTGCs and sets to zero all anomalous fermiongauge vertices δVqq = 0. The next two columns show the effect of letting these anomalou fermion-gauge vertices float, assuming either a MFV or a FU setup respectively.