Studies of dimensionsix EFT effects in vector boson scattering
Abstract
We discuss the implications of dimensionsix operators of the effective field theory framework in the study of vector boson scattering in the \(pp \rightarrow Z Z j j \) channel. We show that operators of dimension six should not be neglected in favour of those of dimension eight. We observe that this process is very sensitive to some of the operators commonly fit using LEP and Higgs data, and that it can be used to improve the bounds on the former. Further we show that other operators than the ones generating anomalous triple and quartic gauge couplings (aTGCs/aQCGs) can have a nonnegligible impact on the total and differential rates and their shapes. For this reason, a correct interpretation of the experimental results can only be achieved by including all the relevant bosonic and fermionic operators; we finally discuss how such an interpretation of experimental measurements can be developed.
1 Introduction
Effective field theories (EFTs) have become extremely popular in the last few years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], proving to be a robust tool for New Physics (NP) searches and BSM physics studies. In general, one can use EFT to find the lowenergy behaviour of a given UV theory (see for example Refs. [37, 38, 39]). Alternatively, one can use EFT in an almost modelindependent way: as a generalised SM extension that can be used to parametrise small deviations observed on experimental measurements of SM observables. The latter is known as the “bottom–up” approach and will be the one used in this work. The main underlying idea of the bottom–up approach is to add higherdimensional operators to the (dimensionfour) Standard Model Lagrangian, in a way that is consistent with the known symmetries: \(SU(2) \times SU(3) \times U(1)\). A variety of relevant early work in this direction can be found in Refs. [40, 41, 42, 43, 44, 45] and some interesting reviews on the topic are Refs. [46, 47, 48]. By adding new higherdimensional terms to the SM Lagrangian, it is possible to parametrise small deviations from the original SM prediction. If such small deviations are found in the experimental data, it is possible to start mapping the space of “New Physics directions”, ruling out some and focussing on others. While few questions remain unanswered in the current picture of fundamental interactions, some seem to be quite far from an answer: How can gravity be put on the same footing as the other fundamental interactions? What are dark matter and dark energy? Other important questions, however, can be tackled at LHC through precision tests of the SM. The most important ones concern the origin of electroweak symmetry breaking (EWSB). The Higgs mechanism [49, 50, 51, 52, 53] has been shown to give a very good description of EWSB, but some details of the latter are still unknown; for example, the fact that the spontaneous symmetry breaking can be realised in linear or nonlinear representations. The answer to this enigma may lie in the gauge couplings, which have only been partially studied at LEP: triple gauge couplings have only been observed in a very concrete energy regime and under a set of assumptions regarding the finalstate radiation whereas the interactions between four gauge bosons will only be observed at the LHC.
To address the last question, the detailed study of the Vector Boson Scattering (VBS) process is essential. This process is characterised at tree level by the exchange of weak gauge bosons between two quarks or a quark and an antiquark, which gives us direct access to both triple and quartic gauge couplings. Since this is a purely electroweak process, it has a relatively small crosssection at LHC where the large QCD backgrounds dominate everywhere. However, the family of VBS processes has very particular experimental signatures: two very energetic forward jets with a big rapidity gap between them. In this work we combine these two interesting fields, by applying EFT techniques to the study of VBS at LHC. In particular, we study the effects of different dimensionsix (\(\mathrm {dim}=6\) ) operators in the total cross section and differential distributions of the purely electroweak contribution to the process \(p p \rightarrow Z Z j j\), more commonly known as VBS(ZZ). We perform a numerical study of the aforementioned quantities, leaving a complete analytical description for future study.
The outline of the paper is as follows: in Sect. 2 we introduce our notation and conventions as well and define of the family of VBS processes. In Sect. 3 we summarise the state of the art, both on theoretical and experimental aspects, and we discuss the issue of anomalous couplings. In Sect. 4.1 we compare some of the published results for dimensionsix operator fits with our predictions for the cross section of this process. In Sects. 4 and 5 we study the impact of different operators of the Warsaw basis on the differential distributions. At first, we focus only on TGC/QGC operators. The full analysis is shown in Sect. 5.1, where we take into account all the Warsawbasis operators, bosonic and fermionic. In Sect. 6 we focus on the main signatures for VBS: dijet observables. Showing that the effects of certain operators (in particular the fourquark ones) can be enhanced on such observables. As anticipated, the family of VBS processes have a relatively small cross section at LHC. The situation for the ZZ final state is particularly dramatic, where the QCD background is very large, with a signal to background ratio of up to 1 / 20 in some phase space regions. For this reason, a rigorous treatment of the process demands also the study of the EFT effects on the corresponding background, which we perform in Sect. 7. Finally, in Sect. 8, we discuss a possible strategy for a global analysis including all the dimensionsix operators relevant to this process.
2 SMEFT: notations and conventions
A general method to construct higherdimensional bases using Hilbert series was proposed in Ref. [56]. In the context of VBS, some subsets of \(\mathrm {dim}=8\) operators affecting quartic gauge couplings have been proposed in Refs. [57, 58].
Other EFT bases There are additional dimensionsix bases, other than the Warsaw basis. It is quite common to use the SILH basis, from Ref. [1] in Higgs phenomenology; however, it is not optimised for multiboson processes. Instead, there is a VBSdedicated basis, typically known as the HISZ basis, from Ref. [59].
Parameter shifts Adding higherdimensional terms to the SM Lagrangian has three consequences: firstly, new vertices appear such as those with four fermions. Secondly, the SM vertices get modified with an additional EFT contribution of the form \(V_\mathrm{SM} = a \cdot g + b \cdot g \cdot c_i/\Lambda ^2\), where g is the SM coupling and \(c_i\) is the Wilson coefficient associated with the \(i\mathrm{th}\) \(\mathrm {dim}=6\) operator. Thirdly, there are shifts on the other SM parameters: the masses, vev, the Weinberg mixing angle and gaugefixing parameters. For a detailed discussion of the parameter shifts and gauge fixing in SMEFT see Refs. [7, 60, 61, 62].

\({\mathcal {O}}_\mathrm{H} = (\Phi ^\dagger \Phi )^3\),

\( {\mathcal {O}}_\mathrm{HD} = (\Phi ^\dagger D_{\mu } \Phi )^* (\Phi ^\dagger D^{\mu } \Phi )\),

\({\mathcal {O}}_{H\Box } = (\Phi ^\dagger \Phi ) \Box (\Phi ^\dagger \Phi )\),
The consequence of the field and parameter redefinitions in Eq. (2.5) is that each and every vertex containing the Higgs field will be dependent on the Wilson coefficients \( \lbrace c_{\mathrm{H} \Box } , c_\mathrm{HD} \rbrace \). This effect is nothing else than a wavefunction renormalisation in the more classical sense.
For this reason, when one studies a concrete process, it is also important to take into account these shifts and not only the EFT effects on single vertices. For example, the operator \({\mathcal {O}}_\mathrm{HB}\) does not directly modify triple or quartic gauge vertices, but it enters the Z field normalisation and hence any vertex containing the former.
In addition to the fields, the parameters \(\lbrace { M}_\mathrm{W}, { M}_\mathrm{z}\), \(\theta _w,\) \( g, G_\mathrm{F} \rbrace \) are also shifted. It is well known that these parameters are not all independent. The relation \({ M}_\mathrm{z}= \frac{{ M}_\mathrm{W}}{\cos (\theta _w)}\), is valid at leading order in the SM, but has to be corrected at NLO. The same holds for the SMEFT, where this and similar relations need to be reexamined (e.g \(\theta _w (g, g')\), \({{ M}_\mathrm{H}} ({\mathrm{G}_\mathrm{F}}, v)\)). For this reason, the input parameter set chosen for a calculation will involve different EFT parameters depending on the choice.
The Wilson coefficients of the effective theory are not observable quantities, and the \(\overline{\mathrm{MS}}\) renormalisation scheme is adopted. The SM masses and the electroweak coupling, on the other hand, can be related to experimental quantities and the onshell renormalisation scheme can be adopted.
2.1 SMEFT amplitudes and cross sections
The leadingorder EFT corresponds to the first term, \({\mathcal {A}}_6^{(1,1)}\). The next term, \({\mathcal {A}}_6^{(1,2)}\), with two insertions at tree level, is tricky, since it is of order \((1/\Lambda ^2)^2\) and the EFT expansion is only order by order renormalisable in the expansion parameter \(1/\Lambda \). For this reason, it is normally not included in the EFT nor in the NLOEFT calculations, unless the corresponding counterterms are available. The term \({\mathcal {A}}_6^{(2,1)}\) accounts for the NLOEFT amplitude, with oneloop–oneinsertion diagrams.
2.2 Offshell effects
Higherdimensional operators are always suppressed by a power of the NP cutoff, \(\Lambda \). This means that the expansion parameter in the perturbative EFT expansion is \(E/\Lambda \), where E is the energy scale of the process under study^{2}, i.e. near the Zpole we can think of the EFT expansion in terms of \({ M}_\mathrm{z}^2/\Lambda ^2\) whereas, away from the peak, the EFT effects are more accurately parametrised by \(p_\mathrm{T}^2/\Lambda ^2\). For this reason, the highenergy regions (tails of the \(p_\mathrm{T}\) distributions) are the ones where the EFT effects are expected to be largest (i.e. \( E_1 / \Lambda \gg E_2/\Lambda \), for \(E_2\) on the pole and \(E_1\) on the tail).
The VBS process is defined by two very energetic jets. This means that the \(p_\mathrm{T}(j)\) and \(m_{jj}\) distributions are privileged kinematic variables as regards where to expect EFT effects.
2.3 VBS: Definition of the process
The family of vector boson scattering processes is very interesting, since it lies at the heart of electroweak symmetry breaking. Some work describing the details of these processes is in Refs. [64, 65, 66, 67, 68]. Unitarity and gauge invariance are conserved in this process thanks to a series of cancellations between Feynman diagrams, and fundamentally thanks to the introduction of the Higgs boson, see Ref. [69, 70]. For these reasons, VBS represents a set of privileged channels for NP studies. For some applications of NP searches to VBS see for example Refs. [71, 72, 73, 74, 75, 76, 77, 78, 79].
There are many possible definitions of the VBS process. Typically, there are substantial differences between the theoretical definitions, in terms of initial and final states, and the experimental definitions, which constrain the phase space of the final states as well. In particular, it is common that the experimental analyses impose certain cuts to try and decouple the vector boson fusion (VBF) process, where a Higgs boson in the schannel is produced from the exchange of weak bosons between a quark and an antiquark. This way, the VBF channel is studied in dedicated Higgs analyses, whereas the VBS channels belong to the multiboson analyses.

\(p_\mathrm{T}(j) > 30\) GeV,

\( m_{jj} > 100\) GeV.

\( \Delta \eta (j_1 j_2) > 2.4\),

\( m_{jj} > 400\) GeV,

\(p_\mathrm{T}(j) > 30\) GeV,

\( \Delta \eta (j_1 j_2) > 2.4\),

\( m_{jj} > 100\) GeV.
As a matter of clarity, in this work we focus only on the process \(p p \rightarrow Z Z j j \) before the onshell decay. The difference between the process \(p p \rightarrow Z Z j j\) followed by the onshell decay \(Z \rightarrow \ell \bar{\ell } \) and the process \(p p \rightarrow \ell \bar{\ell } \, \ell ' \bar{\ell '} j j \) in the aforementioned fiducial region is negligible. In general, when applying multivariate analysis techniques the first definition is preferred, since it populates the phase space in a more effective way.
For a rigorous EFT treatment the same study should be performed including the decays in order to make sure that the difference between the two options is also negligible in SMEFT. It is clear that new operators will come into play, mainly the ones connecting quarks and leptons in the final state; however, intuition and experience tell us that the VBS cuts will most likely remove the bulk of that contribution.
3 Weakboson triple and quartic couplings
3.1 Anomalous couplings
Anomalous gauge couplings were introduced in Ref. [81], at a time when the EWSB mechanism had not been thoroughly tested, and before the Higgs boson was discovered. Such couplings, defined in terms of ad hoc variations on the Lagrangian parameters, might be good in a first approximation, but they present serious theoretical inconsistencies. The main problem is that they violate gauge invariance and unitarity beyond the leading order.
The EFT approach aims to parametrise small deviations from the SM predictions, which are currently being tested with unprecedented precision at LHC. In that regard, a more consistent approach to anomalous couplings is needed. In particular, one that is consistent at nexttoleading order. The SMEFT approach considered in this work, in terms of dimensionsix operators, represents an optimal solution to the anomalous coupling problem: it can be understood as a SM Lagrangian where all the parameters are anomalous, but in a way consistent with the QFT rules.
Numerous works regarding anomalous gauge couplings in SMEFT can be found in the literature. Some or the earliest studies are Refs. [59, 82, 83, 84, 85], and more current ones can be found for example in Refs. [86, 87, 88, 89, 90, 91]. In the upcoming sections we will investigate and discuss the differences between allowing the EFT operators only on the weakcouplings (in the spirit of the anomalous coupling approach) or allowing them to occur anywhere.
3.2 SMEFT for triple and quartic gauge couplings
As part of the legacy of the LEP experiment, it is common in the experimental analysis to study triple gauge couplings (TGCs) in multiboson production channels. For some LEP/LEP2 results, see Refs. [92, 93, 94, 95, 96], and for some LHC analyses, see Refs. [97, 98, 99]. Quartic gauge couplings (QCGs), on the contrary, were more difficult to access at LEP/LEP2 (for example in \(e^+ e^ \rightarrow \gamma \gamma \nu \bar{\nu }\) and \(e^+ e^ \rightarrow \gamma \gamma q \bar{q}\) were studied in Refs. [100, 101]) and not all QGCs where accessible.
This fact makes the QGCs an interesting goal for the LHC experiments, where QGCs are studied in the VBS channels, for example in the analysis of Ref. [80]. Still, it is important to emphasise that this approach is extremely misleading, since it implies identifying a collection of thousands of Feynman diagrams with a single (offshell!) vertex, and more importantly, it implies the assumption that triple and quartic gauge couplings do not originate simultaneously through the EWSB. For this reason, an analysis in terms of cross sections, differential distributions or (pseudo)observables should always be preferred.
At the LEP experiment, the schannel production of Z / Wbosons was very well under control, as well as their decays, since the electroweak radiation could be deconvoluted pretty accurately from the process. This made it possible to treat TGCs as pseudoobservables, which could be measured by the experiments. This is not the case any more at the LHC, and hence one should not aim at “measuring” triple or quartic electroweak couplings.
Pseudoobservables are welldefined theoretical quantities that can be measured in the experiment; a classic example is that of the set of EWPDs. The most promising alternative for LHC physics relies on the study of the residues of Smatrix poles, which are by default gauge invariant quantities. For some work in this direction, see Refs. [102, 103, 104, 105, 106], and the reviews [47, 107].
The impact of \(\mathrm {dim}=6\) operators on triple gauge couplings has been studied in Refs. [88, 89]. The set of \(\mathrm {dim}=8\) operators affecting quartic gauge couplings has been studied in depth in Refs. [57, 58, 108], and another \(\mathrm {dim}=8\) subset, relevant for diboson studies, has been addressed in Ref. [109]. Similar studies to the one presented here, tackling vector boson scattering in the \(\mathrm {dim}=8\) basis have been presented in Refs. [110, 111], as so has work on VBS in the context of the electroweak chiral Lagrangian in Refs. [112, 113]. However, there is no study, to the best of our knowledge, addressing \(\mathrm {dim}=6\) effects in the context of VBS.
3.3 Subsets of operators and gauge invariance
Each of the operators in the Warsaw basis is independently gauge invariant and in principle it is possible in a treelevel study to select a subset of operators without breaking this gauge invariance. However, this situation will not hold beyond tree level, where different operators enter through the \(\mathrm {dim}=6\) counterterms, and the full basis is needed for UVrenormalisation. For a further discussion of the renormalisation of SMEFT see Refs. [7, 8, 10, 60].
Gauge invariance is also broken if the effects of certain \(\mathrm {dim}=6\) or \(\mathrm {dim}=8\) operators are only included on a certain vertex and not in other vertices or wavefunction normalisations. For example, it is gauge invariant to include only \({\mathcal {O}}_W\), \({\mathcal {O}}_\mathrm{HW}\) and \({\mathcal {O}}_\mathrm{HWB}\), neglecting \({\mathcal {O}}_\mathrm{HB}\). But it is not completely rigorous: \({\mathcal {O}}_\mathrm{HB}\) enters every vertex containing a Z field, as shown in Eq. (2.7), and every expression containing the Weinberg mixing angle, and it might enter in different ways depending on the IPS chosen. The same holds for other operators, mainly \({\mathcal {O}}_{\ell \ell }\) and \({\mathcal {O}}_\mathrm{Hl}^{(3)}\), that enter as corrections to \(\mathrm{G}_\mathrm{F}\).
4 EFT for the gauge couplings

\({\mathcal {O}}_\mathrm{W} = \epsilon ^{ijk} W_{\mu }^{i \nu } W_{\nu }^{j \rho } W_{\rho }^{k \mu } \),

\({\mathcal {O}}_\mathrm{HW} = H^{\dagger } H W_{\mu \nu }^I W^{\mu \nu I}\),

\({\mathcal {O}}_\mathrm{HWB} = H^{\dagger } \tau ^I H W_{\mu \nu }^I B^{\mu \nu }\).
In Fig. 2 we see the impact of each of the three operators individually and the sum of them, for four different observables: the invariant mass of the two final Z and that of the two final jets, and the transverse momentum of the leading Z and leading jet. For the numerical values of the coefficients, in this section, we choose the democratic values \(\bar{c}_W = \bar{c}_\mathrm{HW} = \bar{c}_\mathrm{HWB} = 0.06\), which correspond to \({c}_W = {c}_\mathrm{HW} = {c}_\mathrm{HWB} = 1\) with \(\Lambda = 1\)TeV. In Sect. 4.1 we will discuss the case of the available bestfit values.
We observe that the EFT effects on the invariant mass distributions are relatively homogeneous, in particular in the twojet case. The VBS signature is characterised by two very energetic jets and the highenergy phase space is quite well populated. On the contrary, for the ZZ invariant mass we find that at very high energies we reach the limit where the SM production tends to zero, and the EFT effects become sizeable. This effect is, of course, dominated by the Monte Carlo uncertainty, but it points us to a region that should be studied in detail. Such regions where the SM production becomes negligible are also those where the quadratic EFT of Eq. (2.9) will have a dominant role. This will be discussed in Sect. 4.2.
4.1 Comparison with LEP and Higgs bounds
Several works have appeared in the last years, where SMEFT predictions are compared with LEP data (in Refs. [5, 117, 118]) and LHC data (the SILH basis in Refs. [119, 120] and more recently the Warsaw basis in Refs. [121, 122, 123]).
In order to be consistent, a global fit of the full Warsaw basis would be desirable. For this reason, it is necessary to include as many measurements as possible, including fiducial cross sections as well as differential distributions.
In this section we study how the published bestfit values enter the VBS(ZZ) fiducial cross section. In Fig. 3 we show the signal strength \( \mu = \frac{\sigma _\mathrm{EFT}}{\sigma _\mathrm{SM}}\) for the central values given by the profile fit of the operators (black, points) and their \(95\%\) confidence level bounds (red, error bars).^{4}
For the LEP case, we find that the values for some operators are off by a large amount (\(\ge 200\%\)), which means that this channel could be an interesting one to constrain such a fit better. This is not surprising since, as precise as it was, LEP is still a “lowenergy” experiment,^{5} compared with the energies considered here and throughout LHC Run2. Moreover, the treatment of the radiation in LEP raises some doubts on the applicability of such measurements for EFT searches, as discussed in Sect. I.5 of Ref. [47].
4.2 Linear and quadratic contributions

In the regions when the SM prediction becomes very small. In that case the interference term \(\mathrm{SM} \times \mathrm{EFT}_6\) is dominated by the EFT contribution, which is expected to be large. If this is the case, the quadratic term must also be quite large and should be calculated as part of the theoretical uncertainty.

In the cases where the linear \(\mathrm {dim}=8\) interference term is included (\(\mathrm{SM} \times \mathrm{EFT}_8\)), since they are both of the same perturbative order, \({\mathcal {O}}(\Lambda ^4)\).
5 The Warsaw basis in VBS
5.1 EFT for the full process
It is interesting to observe that the bosonic interference is generally positive, while the fermionic one is generally negative. This means that in the case that all the Wilson coefficients would have the same sign, both interferences could extensively cancel, giving rise to a very small SM deviation in the total cross section. For this reason, it is fundamental to define observables and regions where the EFT effects are maximised.
To understand the impact of the full \(\mathrm {dim}=6\) basis, we defined different benchmark scenarios where we study the differential distributions for the most interesting VBS observables.
Benchmarks for bosonic operators. \(\bar{c}_i = c_i \frac{v^2}{\Lambda ^2}\). The number of significant digits used has been reduced in this table for the purpose of clarity. The CPviolating operators \(\lbrace \bar{c}_{\widetilde{\mathrm{W}}}, \bar{c}_{\widetilde{\mathrm{HW}}}, \bar{c}_{\widetilde{\mathrm{HWB}}}, \bar{c}_{\widetilde{\mathrm{HB}}} \rbrace \) have been set to zero
Operator  Benchmark 1  Benchmark 2 

\(\bar{c}_\mathrm{HB}\)  0.0618  \(\) 0.0157 
\(\bar{c}_{H\Box }\)  0.0620  0.109 
\(\bar{c}_\mathrm{Hd}\)  0.0601  0.0872 
\(\bar{c}_\mathrm{HD}\)  0.0685  0.0409 
\(\bar{c}_\mathrm{Hl}^{(3)}\)  0.0761  0.0153 
\(\bar{c}_\mathrm{Hq}^{(1)}\)  0.0600  \(\) 0.0248 
\(\bar{c}_\mathrm{Hq}^{(3)}\)  0.0391  0.0571 
\(\bar{c}_\mathrm{Hu}\)  0.0601  \(\) 0.0481 
\(\bar{c}_\mathrm{HW}\)  0.0576  \(\) 0.0360 
\(\bar{c}_\mathrm{HWB}\)  0.0628  \(\) 0.0402 
\(\bar{c}_\mathrm{W}\)  0.0591  \(\) 0.00507 
\(\mu = \frac{\sigma _\mathrm{EFT}}{\sigma _\mathrm{SM}}\)  0.89  1.14 
5.2 Fourfermion operators
In this section, we repeat the same procedure for purely fermionic operators. An interesting feature of fourfermion operators is that they are always generated at tree level in the UVcompletion, whereas the rest of the dimensionsix operators can be generated either at tree or loop level in the UV theory. This means that the effects of fourfermion operators will often be enhanced by a factor of \(16 \pi ^2\) with respect to the rest of the basis. This is particularly relevant for the case of the study here, since purely gauge operators, those built of three field strength tensors in the \(\mathrm {dim}=6\) case of four field strength tensors in the \(\mathrm {dim}=8\) basis, are always generated from loops in the UVcompletion, and hence always suppressed by \(16 \pi ^2\). For the original work on the PTG/LG classification see Refs. [45]. For more recent discussions see Refs. [11, 130]
In this study, we consider all the fermionic operators. There are 12 operators that dominate the EFT contribution to this process, plus three additional ones which are very colour suppressed. Out of these, only \({\mathcal {O}}_{\ell \ell }\) can be constrained from the available fits, since it enters \(\mathrm{G}_\mathrm{F}\) in the way described in Sect. 2. The rest of the fourfermion operators remain unconstrained, to the best of our knowledge.
Benchmarks for fourfermion operators. \(\bar{c}_i = c_i \frac{v^2}{\Lambda ^2}\). The number of significant digits used has been reduced in this table for the purpose of clarity. We have neglected the operators containing colour structures since they are very suppressed in this process, as shown in Eq. (5.2)
Operator  Benchmark 3  Benchmark 4  Benchmark 5 

\(\bar{c}_\mathrm{dd}\)  \(\) 0.032  0.061  0.023 
\(\bar{c}_\mathrm{dd}^{(1)}\)  0.0077  0.061  0.092 
\(\bar{c}_{\ell \ell }^{(1)}\)  \(\) 0.042  0.039  0.0036 
\(\bar{c}_\mathrm{qd}^{(1)}\)  \(\) 0.033  0.060  0.076 
\(\bar{c}_\mathrm{qq}^{(1)}\)  0.0099  0.050  \(\) 0.042 
\(\bar{c}_\mathrm{qq}^{(11)}\)  0.024  0.047  \(\) 0.043 
\(\bar{c}_\mathrm{qq}^{(3)}\)  0.023  \(\) 0.017  \(\) 0.042 
\(\bar{c}_\mathrm{qq}^{(31)}\)  \(\) 0.038  0.0031  0.094 
\(\bar{c}_\mathrm{qu}^{(1)}\)  0.051  0.060  \(\) 0.0020 
\(\bar{c}_\mathrm{ud}^{(1)}\)  0.071  0.060  \(\) 0.084 
\(\bar{c}_\mathrm{uu}\)  \(\) 0.099  0.060  \(\) 0.037 
\(\bar{c}_\mathrm{uu}^{(1)}\)  \(\) 0.087  0.060  \(\) 0.029 
\(\mu = \frac{\sigma _\mathrm{EFT}}{\sigma _\mathrm{SM}}\)  0.83  1.15  0.80 
6 Dijet observables
7 EFT in the background
The main background at LHC for the previously studied process is the QCD induced process \(p p \rightarrow z z j j \). This has a very large cross section compared with the VBS one and it is mainly discriminated from the signal thanks to the jet signature of the latter. This background was analysed in LHC in Refs. [97, 133].
Moreover, in the available VBS(ZZ) analysis, the S/B discrimination was only achieved by means of a boosted decision tree (BDT) and a matrix element (ME) discriminator, described in Refs. [80, 134].
Given this situation, it would not be sensible to add the EFT effects in the signal and not in the much larger background. In the following we show a preliminary study of the EFT effects in the background and we discuss which regions and observables are best for the observation of EFT effects (Fig. 9).
7.1 Characterisation of the background
This means that the EFT effects in the background should always be well studied and understood. Else, an effect like this, background enhancement and signal reduction, could lead us to missing some interesting EFT effects if we only look at signals and total rates.
8 General strategy for EFT studies at LHC
In this work we have shown that at least 11 bosonic and 12 fermionic dimensionsix operators enter the VBS(ZZ) signal cross section. If we try to isolate the effects of each of these operators using a single cross section measurement, the solutions to this system define a 22dimensional manifold. In order to further constrain the space of solutions it is necessary to add further experimental measurements.
In this work we have proposed different VBS observables^{7}: invariant mass of the two gauge bosons, invariant mass and rapidity of the dijet system, transverse momentum of the leading gauge boson and jet and the rapidity separation between jets. As we have seen throughout Sect. 5, the EFT scales differently in different regions of phase space (with the highenergy “tails” being privileged). This means that a differential measurement with n bins could be used to add n equations to the aforementioned 23 variable system, reducing the number of directions in the solution hyperplane.
It is always possible to add different measurements (from LEP, Higgs physics, etc.), in order to reduce the number of unknowns in the system of equations. But this should be dealt with taking great care: a bound set for a certain Wilson coefficient is only valid in the energy regime where the calculation is performed. In order to know the value of this coefficient at a different energy scale, the renormalisation group evolution of the coefficient has to be calculated. In that regard, it is safer to include more observables and more bins for signal and background measurements than to mix VBSsignal, Higgssignal and LEPsignal measurements.
Ideally, gauge boson polarisation measurements, not studied in detail here, could also be useful. Such polarisation studies happen to be very interesting already at the SM level, as pointed out in recent work like Refs. [78, 136]. Furthermore, the study of the parton shower effects on the EFT distributions, might also shed new light on the problem. There are very few references in this direction so far, for example Ref. [123]. Last but not least, the study of the backgrounds, as shown in Sect. 7, is also necessary for a correct determination of the EFT effects.
9 Conclusions
At the current level of precision of LHC measurements, and the high level of agreement of these measurements with the SM predictions, it is advisable to perform any EFT analysis with a great deal of accuracy. For this purpose EFT operators cannot be studied on a casebycase basis, and a global study of the set of dimensionsix operators is necessary in the first place. In a second stage, dimensioneight as well as NLO and quadratic dimensionsix effects need to be studied in order to improve the EFT theoretical uncertainties.
Footnotes
 1.
 2.
This interpretation is consistent with the fact that the EFT coefficients \(c_i\) run with the energy scale, just as every other Lagrangian parameter.
 3.
The analysis can easily be extended to the CPodd case, by adding \({\mathcal {O}}_{\widetilde{\mathrm{W}}}\), \({\mathcal {O}}_{\widetilde{\mathrm{HW}}}\) and \({\mathcal {O}}_{\widetilde{\mathrm{HWB}}}\).
 4.
The data are taken from the aforementioned papers. For the LEP fit we took the values corresponding to the \({ M}_\mathrm{W}\) IPS and to a \(1\%\) theoretical uncertainty. For the Higgs fit we took the values in Table 4 of Ref. [122], but it was not possible to clarify such details (IPS and th. uncertainty) with the authors, these could be the origin of some of the large interference terms found.
 5.
An important step of any EFT calculation requires the matching of the EFT calculation, at a low energy scale \(E_0\), with the “new physics” scale \(\Lambda \), to account for the running of the EFT coupling.
 6.
The operator \({\mathcal {O}}_\mathrm{HG}\) is very well understood through the gluon–gluon fusion process, and \({\mathcal {O}}_{\widetilde{\mathrm{HG}}}\) is CPodd.
 7.
We focus on the ZZ channel here, but this logic applies equally well to the other VBS channels.
Notes
Acknowledgements
We want to thank Giampiero Passarino for relevant comments on the manuscript, and Roberto Covarelli for relevant comments regarding the experimental setup. We gratefully acknowledge Michael Trott for providing us with the likelihood functions for the fits presented in Refs. [117, 118] as well as Ilaria Brivio for technical support with the SMEFTsim package, very interesting comments and discussions, and relevant input at the beginning of this project. We also want to thank Pietro Govoni for hosting us at the initial stage of this project, supported by a STSM Grant from COST Action CA16108, as well as the other members of the Action for interesting discussions.
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