# Same-sign WW scattering at the LHC: can we discover BSM effects before discovering new states?

## Abstract

It is possible that measurements of vector boson scattering (VBS) processes at the LHC will reveal disagreement with Standard Model predictions, but no new particles will be observed directly. The task is then to learn as much as possible about the new physics from a VBS analysis carried within the framework of the Effective Field Theory (EFT). In this paper we discuss issues related to the correct usage of the EFT when the *WW* invariant mass is not directly accessible experimentally, as in purely leptonic *W* decay channels. Strategies for future data analyses in case such scenario indeed occurs are proposed.

## 1 Introduction and strategy

Searches for deviations from Standard Model (SM) predictions in processes involving interactions between known particles are a well established technique to study possible contributions from Beyond the Standard Model (BSM) physics. In this paper we address the question how much we can learn about the scale of new physics and its strength using the Effective Field Theory (EFT) approach to \(W^+W^+\) scattering if a statistically significant deviation from the SM predictions is observed in the expected LHC data for the process \(pp \rightarrow 2 jets + W^+W^+ \). Our specific focus is on the proper use of the EFT in its range of validity. With this in mind, we discuss the practical usefulness of the EFT language to describe vector boson scattering (VBS) data and whether or not this can indeed be the right framework to observe the first hints of new physics at the LHC.

*C*’s are some “coupling constants” and \(\Lambda \)’s are the decoupled new mass scales. The mass scale is a feature of the UV completion of the full theory and thus is assumed common to all the coefficients

*f*’s and not the scale \(\Lambda \) itself. Neither \(\Lambda \) nor the

*C*’s are known without referring to specific UV complete models. Even for \(E<<\Lambda \) a simple counting of powers of \(E/\Lambda \) can be misleading as far as the contribution of various operators to a given process is concerned. The latter depends also on the relative magnitude of the couplings

*C*, e.g., \(C_i^{(6)}\) versus \(C_i^{(8)}\) and/or within each of those sets of operators, separately, [1, 2, 3, 4], as well as on the interference patterns in various amplitudes calculated from the Lagrangian Eq. (1.1) [5].

For instance, the contribution of dimension-6 (\(\hbox {D}=6\)) operators to a given process can be suppressed compared to dimension-8 (\(\hbox {D}=8\)) operators contrary to a naive \((E/\Lambda )\) power counting [3, 4, 5, 6] or, vice versa, the [\(\hbox {SM}\ \times \hbox {D}=8\)] interference contribution can be subleading with respect to the \([D=6]^2\) one [2, 4, 7]. Clearly, the assumption about the choice of operators in the truncation in Eq. (1.1) used to analyze a process of our interest introduces a strong model dependent aspect of that analysis: one is implicitly assuming that there exist a class of UV complete models such that the chosen truncation is a good approximation. It is convenient to introduce the concept of EFT “models” defined by the choice of operators \({\mathcal {O}}_i\) and the values of \(f_i\). The question of this paper is then about the discovery potential at the LHC for BSM physics described by various EFT “models”.

The crucial question is what the range of validity can be of a given EFT “model”. There is no precise answer to this question unless one starts with a specific theory and derives Eq. (1.1) by decoupling the new degrees of freedom. However, in addition to the obvious constraint that the EFT approach can be valid only for the energy scale \(E<\Lambda \) (unfortunately with unknown value of \(\Lambda \)), for theoretical consistency the partial wave amplitudes should satisfy the perturbative unitarity condition. The latter requirement translates into the condition \(E^2<\Lambda ^2\le s^U\), where \(s^U\equiv s^U(f_i)\) is the perturbative partial wave unitarity bound as a function of the chosen operators and the values of the coefficients \(f_i\)’s. Thus, the value of \(\Lambda ^2_{max}=s^U\) gives the upper bound on the validity of the EFT based “model”. Since the magnitude of the expected (or observed) experimental effects also depends on the same \(f_i\), one has a frame for a consistent use of the EFT “model” to describe the data once they are available. For a BSM discovery in the EFT framework, proper usage of the “model” is a vital issue. It makes no physical sense to extend the EFT “model” beyond its range of applicability, set by the condition \(E<\Lambda \). We shall illustrate this logic in more detail in the following.

*choosing*different EFT “models”: such limits are valid only under the assumption that just one chosen operator dominates BSM effects in the studied process in the available energy range. In this paper we will consider only variations of single dimension-8 operators.

^{1}However, the strategy we present can be extended to the case of many operators at a time, including dimension-6 (keeping in mind that varying more than one operator substantially complicates the analysis). For a given EFT “model”

*l*and \(l'\) stand for any combination of electrons and muons. The process depends on the \(W^+W^+\) scattering amplitude (the gauge bosons can of course be virtual). The EFT “models” can be maximally valid up to certain invariant mass \(M=\sqrt{s}\) of the \(W^+W^+\) system

*i*, the sum runs over partons in the initial (

*ij*) and final (

*kl*) states and over helicities, the amplitude \({\mathcal {M}}\) is for the parton level process \(ij\rightarrow klW^+W^+\) and \(d\Omega \) denotes the final state phase space integration. The special role of the distribution \(\frac{d\sigma }{dM}\) follows from the fact that it is straightforward to impose the cutoff \(M\le \Lambda \), Eq. (1.5), for the

*WW*scattering amplitude. The differential cross section, \(\frac{d\sigma }{dM}\), is therefore a very sensitive and straightforward test of new physics defined by a given EFT “model”. Unfortunately, the \(W^+W^+\) invariant mass in the purely leptonic

*W*decay channel is not directly accessible experimentally and one has to investigate various experimental distributions of the charged particles. The problem here is that the kinematic range of those distributions is not related to the EFT “model” validity cutoff \(M<\Lambda \) and if \(\Lambda < M_{max}\), where \(M_{max}\) is the kinematic limit accessible at the LHC for the

*WW*system, there is necessarily also a contribution to those distributions from the region \(\Lambda< M < M_{max}\). The question is then: in case a deviation from SM predictions is indeed observed, how to verify a “model” defined by a single higher-dimension operator \({\mathcal {O}}_i^{(k)}\) and a given value of \(f_i\) by fitting it to a set of experimental distributions \(D_i\) and in what range of \(f_i\) such a fit is really meaningful [7]. Before we address this question, it is in order to comment on the perturbative partial wave unitarity constraint.

- 1.
For a given EFT “model”, the unitarity bound is very different for the \(J=0\) partial wave of different helicity amplitudes and depends on their individual energy dependence (some of them remain even constant and never violate unitarity, see Appendix). Our \(M^U\) has to be taken as the

*lowest*unitarity bound, universally for all helicity amplitudes, because it is the lowest bound that determines the scale \(\Lambda _{max}\). More precisely, one should take the value obtained from diagonalization of the matrix of the \(J=0\) partial waves in the helicity space. - 2.
Correct assessment of the EFT “model” validity range in the \(W^+W^+\) scattering process requires also consideration of the \(W^+W^-\) scattering amplitudes which by construction probe the same couplings and are sensitive to exactly the same operators. For most higher dimension operators, this actually significantly reduces their range of validity in \(W^+W^+\) analyses. Conversely, the

*WZ*and*ZZ*processes can be assumed to contain uknown contributions from additional operators which adjust the value of \(\Lambda \) consistently. - 3.
It is interesting to note that for the \(f_i\) values of practical interest the deviations from SM predictions in the total cross sections become sizable only in a narrow range of energies just below the value of \(M^U\), where the \(|A_{BSM}|^2\) term in Eq. (1.3) takes over. However, for most dimension-8 operators the contribution of the interference term is not completely negligible (see Appendix for details). Even if deviations from the SM are dominated by the helicity combinations that reach the unitarity bound first, the total unpolarized cross sections up to \(M=M^{U}\) get important contributions also from amplitudes which are still far from their own unitarity limits.

*any*way that preserves unitarity of the amplitudes, i.e., the contribution from this region is sufficiently suppressed kinematically by parton distributions. This requirement is of course impossible to impose in practice, but for a rough quantitative estimate of the magnitude of this contribution, one can assume that all the helicity amplitudes above \(\Lambda \) remain constant at their respective values they reach at \(\Lambda \), and that \(\Lambda \) is common to all the helicity amplitudes. For \(\Lambda = \Lambda _{max}\), this prescription regularizes the helicity amplitudes that violate unitarity at \(M^U\) and also properly accounts for the contributions of the helicity amplitudes that remain constant with energy. It gives a reasonable approximation to the total unpolarized cross sections for \(M>M^U\), at least after some averaging over

*M*. More elaborated regularization techniques can also be checked here. The full contribution to a given distribution \(D_i\) is then taken as

*f*to obtain the required signal statistical significance. It can be derived based on Eq. (1.9) [or Eq. (1.8)]. On the other hand, description in the EFT language imposes some maximum value of

*f*such that signal estimates computed from Eqs. (1.8) and (1.9) remain statistically consistent. Large difference between the two computations implies significant sensitivity to the region above \(\Lambda \). It impedes a meaningful data description in the EFT language and also suggests we are more likely to observe the new physics directly.

*f*values, bounded from two sides, for which BSM discovery and correct EFT description are both plausible. In the more general case when \(\Lambda <M^U\), i.e., new physics states may appear before our EFT “model” reaches its unitarity limit, respective limits on

*f*depend on the actual value of \(\Lambda \). We thus obtain a 2-dimensional region in the plane \((\Lambda , f_i)\), which is shown in the cartoon plot in . 1. This region is bounded from above by the unitarity bound \(M^U(f_i)\) (solid blue curve), from the left by the signal significance criterion (dashed black curve) and from the right by the EFT consistency criterion (dotted black curve). The EFT could be the right framework to search for BSM physics as long as these three criteria do not mutually exclude each other, i.e., graphically, the “triangle” shown in our cartoon plot is not empty. In Sect. 3 we will verify whether such “triangles” indeed exist for the individual dimension-8 operators.

- 1.
From collected data measure a distribution \(D_i\) (possibly in more than one dimension) that offers the highest sensitivity to the studied operator(s).

- 2.
If deviations from the SM are indeed observed,

^{2}fit particular values of \((\Lambda \le M^U, f_i)\) based on EFT simulated templates in which the contribution from the region \(M > \Lambda \) is taken into account according to Eq. (1.9) or using some more elaborated regularization methods. - 3.
Fixing \(f_i\) and \(\Lambda \) to the fit values, recalculate the \(D_i\) template so that the region \(M > \Lambda \) is populated only by the SM contribution Eq. (1.8).

- 4.
Check statistical consistency between the original simulated \(D_i\) template and the one based on Eq. (1.8).

- 5.
Physics conclusions from the obtained \((\Lambda , f_i)\) values can only be drawn if such consistency is found. In addition, stability of the result against different regularization methods provides a measure of uncertainty of the procedure – too much sensitivity to the region above \(\Lambda \) means the procedure is destined to fail and so the physical conclusion is that data cannot be described with the studied operator.

## 2 Preliminary technicalities

Higgs and triple gauge couplings can be accessed experimentally via other processes, namely Higgs physics and diboson production which is most sensitive to aTGC. They are presently known to agree with the SM within a few per cent (For a combined analysis of LEP and Run I LHC data, see e.g. [14]), which translates into stringent limits on the dimension-6 operators.

*WWWW*vertex:

The same-sign \(pp\rightarrow W^+W^+ jj\) production has been already observed during Run I of the LHC [8, 9, 10] and confirmed by a recent measurement of the CMS Collaboration at 13 TeV Run II [11]. Also, pioneering measurements of the \(ZW^\pm jj\) [15] and *ZZjj* [11] processes exist. They all place experimental limits on the relevant dimension-8 operators. However, most presently obtained limits involve unitarity violation within the measured kinematic range, leading to problems in physical interpretation and even comparison of the different analyses.

Our goal is to investigate the discovery potential at the High Luminosity LHC (HL-LHC) of the BSM physics effectively described by EFT “models” with single dimension-8 operators at a time, with proper attention paid to the regions of validity of such models, as described in Sect. 1.

## 3 Results of simulations

For the following analysis dedicated event samples of the process \(pp \rightarrow jj\mu ^+\mu ^+\nu \nu \) at 14 TeV were generated at LO using the MadGraph5_aMC@NLO v5.2.2.3 generator [17], with the appropriate UFO files containing additional vertices involving the desired dimension-8 operators. For each dimension-8 operator a sample of at least 500,000 events within a phase space consistent with a VBS-like topology (defined below) was generated. A preselected arbitrary value of the relevant *f* coefficient (from now on, \(f \equiv f_i\) with \(i = S0, S1, T0, T1, T2, M0, M1, M6, M7\)) was assumed at each generation; different *f* values were obtained by applying weights to generated events, using the reweight command in MadGraph. The value *f*=0 represents the Standard Model predictions for each study. The Pythia package v6.4.1.9 [20] was used for hadronization as well as initial and final state radiation processes. No detector was simulated. Cross sections at the output of MadGraph were multiplied by a factor 4 to account for all the lepton (electron and/or muon) combinations in the final state.

In this analysis, the Standard Model process \(pp \rightarrow jjl^+l^+\nu \nu \) is treated as the irreducible background, while signal is defined as the enhancement (which may be positive or negative in particular cases) of the event yield in the presence of a given dimension-8 operator relative to the Standard Model prediction. No reducible backgrounds were simulated, as they are known to be strongly detector dependent. For this reason, results presented here should be treated mainly as a demonstration of our strategy rather than as a precise determination of numerical values. For more realistic results this analysis should be repeated with full detector simulation for each of the LHC experiments separately.

*f*. The \(M_{WW} > \Lambda \) tail of the distribution is then assumed identical as in the Standard Model case. Second, using Eq. (1.9) which accounts for an additional BSM contribution coming from the region \(M_{WW} > \Lambda \). The latter is estimated under the assumption that helicity amplitudes remain constant above this limit, as discussed in Sect. 1. For the case when \(\Lambda \) is equal to the unitarity limit, this corresponds to unitarity saturation.

For each *f* value of every dimension-8 operator, signal significance is assessed by studying the distributions of a large number of kinematic variables. We only considered one-dimensional distributions of single variables. Each distribution was divided into 10 bins, arranged so that the Standard Model prediction in each bin is never lower than 2 events. Overflows were always included in the respective highest bins. Ultimately, each distribution had the form of 10 numbers, that represent the expected event yields normalized to a total integrated luminosity of \(3\ \hbox {ab}^{-1}\), each calculated in three different versions: \(N^{SM}_i\) for the Standard Model case, \(N^{EFT}_i\) from applying Eq. (1.8), and \(N^{BSM}_i\) from applying Eq. (1.9) (here subscript *i* runs over the bins). In this analysis, Eq. (1.9) was implemented by applying additional weights to events above \(M_{WW} = \Lambda \) in the original non-regularized samples generated by MadGraph. For the dimension-8 operators, this weight was equal to \((\Lambda /M_{WW})^4\). The choice of the power in the exponent takes into account that the non-regularized total cross section for *WW* scattering grows less steeply around \(M_{WW} = \Lambda \) than its asymptotic behavior \(\sim s^3\), which is valid in the limit \(M_{WW} \rightarrow \infty \). This follows from the observation that unitarity is first violated much before the cross section gets dominated by its \(\sim s^3\) term, as shown in the Appendix. The applied procedure is supposed to ensure that the total *WW* scattering cross section after regularization behaves like 1 / *s* for \(M_{WW} > \Lambda \), and so it approximates the principle of constant amplitude (Sect. 1), at least after some averaging over the individual helicity combinations. Examples of simulated distributions are shown in Fig. 2.

*f*are shown in Fig. 3 with dashed curves. Consistency of the EFT description is determined by requiring a small difference between the respective predictions from Eqs. (1.8) and (1.9). An additional \(\chi ^2_{add}\) is computed based on the comparison of the respective distributions of \(N^{EFT}_i\) and \(N^{BSM}_i\):

*f*is shown in Fig. 3 as dotted curves. These considerations consequently translate into effective upper limits on the value of

*f*for each operator.

For each dimension-8 operator we took the distribution that produced the highest \(\chi ^2\) among the considered variables. The most sensitive variables we found to be \(R_{p_T} \equiv p_T^{~l1}p_T^{~l2}/(p_T^{~j1}p_T^{~j2})\) [21] for \({\mathcal {O}}_{S0}\) and \({\mathcal {O}}_{S1}\), and \(M_{o1} \equiv \sqrt{(|\mathbf {p}_T^{~l1}|+|\mathbf {p}_T^{~l2}| +|\mathbf {p}_T^{~miss}|)^2 - (\mathbf {p}_T^{~l1}+\mathbf {p}_T^{~l2}+\mathbf {p}_T^{~miss})^2}\) [22] for the remaining operators (for some of them, \(M_{ll}\) would give almost identical results as \(M_{o1}\), but usually this was not the case).

Unitarity limits were computed using the VBFNLO [18, 19] calculator v1.3.0, after applying appropriate conversion factors to the input values of the Wilson coeeficients, so to make it suitable to the MadGraph 5 convention. We used the respective values from T-matrix diagonalization, considering both \(W^+W^+\) and \(W^+W^-\) channels, and taking always the lower value of the two. For the operators we consider here, unitarity limits are lower for \(W^+W^-\) than for \(W^+W^+\) except for \(f_{S0}\) (both positive and negative) and negative \(f_{T1}\).

*f*for each dimension-8 operator, for positive and negative

*f*values, estimated for the HL-LHC with an integrated luminosity of \(3\ \hbox {ab}^{-1}\), are read out directly from graphs such as Fig. 3 and listed below in Table 1. These limits define the (continous) sets of testable EFT “models” based on the choice of single dimension-8 operators.

Estimated lower limits for BSM signal significance and upper limits for EFT consistency for each dimension-8 operator (positive and negative *f* values), for the case when \(\Lambda \) is equal to the unitarity bound, in the \(W^+W^+\) scattering process at the LHC with \(3\ \hbox {ab}^{-1}\)

Coeff. | Lower limit | Upper limit | Coeff. | Lower limit | Upper limit |
---|---|---|---|---|---|

(\(\hbox {TeV}^{-4}\)) | (\(\hbox {TeV}^{-4}\)) | (\(\hbox {TeV}^{-4}\)) | (\(\hbox {TeV}^{-4}\)) | ||

\(f_{S0}\) | 1.3 | 2.0 | \(-f_{S0}\) | 1.2 | 2.0 |

\(f_{S1}\) | 8.0 | 6.5 | \(-f_{S1}\) | 5.5 | 6.0 |

\(f_{T0}\) | 0.08 | 0.13 | \(-f_{T0}\) | 0.05 | 0.12 |

\(f_{T1}\) | 0.03 | 0.06 | \(-f_{T1}\) | 0.03 | 0.06 |

\(f_{T2}\) | 0.20 | 0.25 | \(-f_{T2}\) | 0.10 | 0.20 |

\(f_{M0}\) | 1.0 | 1.2 | \(-f_{M0}\) | 1.0 | 1.2 |

\(f_{M1}\) | 1.0 | 1.9 | \(-f_{M1}\) | 0.9 | 1.8 |

\(f_{M6}\) | 2.0 | 2.4 | \(-f_{M6}\) | 2.0 | 2.4 |

\(f_{M7}\) | 1.1 | 2.8 | \(-f_{M7}\) | 1.3 | 2.8 |

The fact that the obtained lower limits are more optimistic than those from several earlier studies (see, e.g., Ref. [23]) reflects our lack of detector simulation and reducible background treatment, but may be partly due to the use of the most sensitive kinematic variables. It must be stressed, nonetheless, that both these factors affect all lower and upper limits likewise, so their relative positions with respect to each other are unlikely to change much.

Allowing that the scale of new physics \(\Lambda \) may be lower than the actual unitarity bound results in 2-dimensional limits in the \((f,\Lambda )\) plane. Usually this means further reduction of the allowed *f* ranges for lower \(\Lambda \) values and the resulting regions take the form of an irregular triangle. Respective results for all the dimension-8 operators are depicted in Figs. 4 and 5. It is interesting to note that in many cases this puts an effective lower limit on \(\Lambda \) itself, in addition to the upper limit derived from the unitarity condition. In particular, the adopted criteria bound the value of \(\Lambda \) to being above \(\sim 2\ \hbox {TeV}\) for the \({\mathcal {O}}_M\) operators as well as for \({\mathcal {O}}_{S0}\). The \({\mathcal {O}}_T\) operators still allow a wider range of \(\Lambda \). Unfortunately, there is little we can learn from fitting \(f_{S1}\), since signal observability requires very low \(\Lambda \) values, for which the new physics could probably be detected directly.

*k*is the dimensionality of the operator that defines the EFT “model”. In models with one BSM scale and one BSM coupling constant \(\sqrt{C}\) has the interpretation of the coupling constant [1]. The values of \(C_{max}\) are to a good approximation independent of

*f*(see Fig. 6) and, being generally in the range (\(\sqrt{4\pi }, 4\pi \)), reflect the approach to a strongly interacting regime in an underlying (unknown) UV complete theory. The EFT discovery regions depicted in Figs. 4 and 5 have further interesting implications for the couplings

*C*. For a fixed

*f*, the unitarity bound \(\Lambda ^2<s^U\) implies that \(C<C_{max}=f(M^U)^4\), whereas the lower bound on \(\Lambda \) that comes from the combination of the signal significance and EFT consistency criteria gives us \(C>\Lambda _{min}^ 4f\). Thus, a given range \((\Lambda _{min}, \Lambda _{max})\) corresponds to a range of values of the couplings

*C*, so that we could not only discover an indirect sign of BSM physics, but also learn something about the nature of the complete theory, whether it is strongly or weakly interacting. In particular, for the following operators: \({\mathcal {O}}_{S0}\), \({\mathcal {O}}_{M0}\), \({\mathcal {O}}_{M1}\), \({\mathcal {O}}_{M6}\) and \({\mathcal {O}}_{M7}\), only models with

*C*being close to the strong interaction limit will be experimentally testable, while a wider range of

*C*may be testable for \({\mathcal {O}}_{T0}\), \({\mathcal {O}}_{T1}\) and \({\mathcal {O}}_{T2}\).

## 4 Conclusions and outlook

In this paper we have analyzed the prospects for discovering physics beyond the SM at the HL-LHC in the EFT framework applied to the VBS amplitudes, in the process \(pp\rightarrow W^+W^+jj\). We have introduced the concept of EFT “models” defined by the choice of higher dimension operators and values of the Wilson coefficients and analyzed “models” based on single dimension-8 operators at a time. We emphasize the role of the invariant mass \(M_{WW}\) whose distribution directly relates to the intrinsic range of validity of the EFT approach, \(M_{WW} < \Lambda \le M^U\), and the importance to tackle this issue correctly in data analysis in order to study the underlying BSM physics. While this is relatively simple (in principle) for final states where \(M_{WW}\) can be determined on an event-by-event basis, the value of \(M_{WW}\) is unfortunately not available in leptonic *W* decays. We argue that usage of EFT “models” in the analysis of purely leptonic *W* decay channels requires bounding the possible contribution from the region \(M_{WW} > \Lambda \), no longer described by the “model”, and ensuring it does not significantly distort the measured distributions compared to what they would have looked from the region of EFT validity alone.

We propose a data analysis strategy to satisfy the above requirements and verify in what ranges of the relevant Wilson coefficients such strategy can be successfully applied in a future analysis of the HL-LHC data. We find that, with a possible exception of \({\mathcal {O}}_{S1}\), all dimension-8 operators which affect the *WWWW* quartic coupling have regions where a \(5 \sigma \ \hbox {BSM}\) signal can be observed at HL-LHC with \(3\ \hbox {ab}^{-1}\) of data, while data could be satisfactorily described using the EFT approach.

From such analysis it may be possible to learn something about the underlying UV completion of the full theory. Successful determination of a given *f* value, using a procedure that respects the EFT restricted range of applicability, will put non-trivial bounds on the value of \(\Lambda \) and consequently, the BSM coupling *C*. These bounds are rather weak for \({\mathcal {O}}_{T0}\), \({\mathcal {O}}_{T1}\) and \({\mathcal {O}}_{T2}\) operators, but potentially stronger for \({\mathcal {O}}_{M0}\), \({\mathcal {O}}_{M1}\), \({\mathcal {O}}_{M6}\) and \({\mathcal {O}}_{M7}\). In particular, applicability of the EFT in terms of these operators already requires \(\Lambda \ge 2\ \hbox {TeV}\), while stringent upper limits arise from the unitarity condition. Because of relatively low sensitivity to \(f_{S0}\) and \(f_{S1}\), it will unfortunately be hard to learn much about \(W_LW_L\) physics using the EFT approach with dimension-8 operators.

It must be stressed that in this analysis we have only considered single dimension-8 operators at a time. Allowing non-zero values of more than one *f* at a time provides much more felixibility as far as the value of \(\Lambda \) is concerned, especially for those operators whose individual unitarity limits are driven by helicity combinations which contribute little to the total cross section. Consequently, regions of BSM observability and EFT consistency can only be larger than what we found here. Study of VBS processes in the EFT language can be the right way to look for new physics and should gain special attention in case the LHC fails to observe new physics states directly.

Consideration of other VBS processes and *W* decay channels may significantly improve the situation. In particular, the semileptonic decays, where one \(W^+\) decays leptonically and the other \(W^+\) into hadrons, have never been studied in VBS analyses because of their more complicated jet combinatorics and consequently much higher background. Progress in the implementation of *W*-jet tagging techniques based on jet substructure algorithms may render these channels interesting again. However, they are presently faced with two other experimental challenges. One is the precision of the \(M_{WW}\) determination which relies on the missing-\(E_T\) measurement resolution. The other one is poor control over the sign over the hadronic *W*. The advantages would be substantial. If \(M_{WW}\) can be reconstructed with reasonable accuracy, it is straightforward to fit *f* and \(\Lambda \) to the measured distribution in an EFT-consistent way even for arbitrarily large *f*. Existence of a high-\(M_{WW}\) tail above \(\Lambda \) is then not a problem, but a bonus, as it may give us additional hints about the BSM physics. Finally, because of the invariant mass issue, the *ZZ* scattering channel, despite its lowest cross section, may ultimately prove to be the process from which we can learn the most about BSM in case the LHC fails to discover new physics directly.

## Footnotes

- 1.
For a physical justification of omitting dimension-6 operators see Sect. 2.

- 2.
We do not discuss in this paper the bounds on the Wilson coefficients obtained from the data analysis when no statistically significant signal on new physics is observed. Such an analysis requires a separate discussion, although it will be also influenced by the results of this paper.

## Notes

### Acknowledgements

The work of SP is partially supported by the National Science Centre, Poland, under research grants DEC-2014/15/B/ST2/02157, DEC-2015/18/M/ST2/00054 and DEC-2016/23/G/ST2/04301, and that of JK by the Polish National Science Center HARMONIA project under contract UMO-2015/18/M/ST2/00518 (2016-2019). JR is supported in part by the National Science Center, Poland, under research grant DEC2015/19/B/ST2/02848. ST is supported by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.

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