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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
In this work, the bottom–up approach to EFT is used. The SM Lagrangian is extended with higherdimensional operators consistent with the known SM symmetries. Furthermore, we assume a linear representation for the physical Higgs field, in the form an \(\mathrm{SU}(2)\) doublet. Such a theory is commonly known as SMEFT:
At dimension five, \(\mathrm {dim}=5\), there is only one possible operator, from Ref. [54], which does not enter the process studied in this work. At \(\mathrm {dim}=6\) , the complete basis has 59 operators in the flavour universal case and 2499 in the most general case. In this work we will use a parametrisation of the former, commonly known as the Warsaw basis [55].
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].
The easiest example to understand parameter shifts is that of the Higgs field: if we add the Warsawbasis operators to the SM Lagrangian, the Higgs part of the Lagrangian becomes
where:

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

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

\({\mathcal {O}}_{H\Box } = (\Phi ^\dagger \Phi ) \Box (\Phi ^\dagger \Phi )\),
and \(\Phi \) is the Higgs doublet,
in the Feynman and unitary gauges, respectively. By expanding Eq. (2.2) we see that the kinetic term is modified as well as the potential (i.e. its minimum gets shifted). To restore the correct vacuum expectation value and canonical normalisation of the kinetic term, the corresponding SM parameter and field have to be redefined as
After some simple algebra one finds
As is customary in the literature, we redefine the Wilson coefficients as \( \bar{c}_i = \frac{v^2}{\Lambda ^2} c_i\).
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.
The same phenomenon occurs for all the other fields and parameters. The case of the EW sector has to be handled with great care, especially when working on the Feynman gauge since new linear transitions between gauge and Goldstone bosons appear. The gauge sector of the Lagrangian becomes
As a consequence, the gauge fields get shifted in the same fashion as shown for the Higgs field. The neutral gauge boson for example^{Footnote 1}:
where \(\theta _w\) is the Weinberg mixing angle.
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.
Importance of the input parameter set (IPS) It is important to recall that different IPSs lead to different predictions, already at tree level. While the “\(\alpha \)scheme”: \(\{ { M}_\mathrm{z}, \alpha , \mathrm{G}_\mathrm{F}\}\) might be convenient at lower energies, or when using experimental measurements of EWPD, for the energy scale of our process (\(E \approx 2 { M}_\mathrm{z}\)) the “\({ M}_\mathrm{W}\)scheme”, \(\{ { M}_\mathrm{z}, { M}_\mathrm{W}, \mathrm{G}_\mathrm{F}\}\), is a better choice. The modifications to the vev are shown in Eq. (2.5), and analogously, \(\mathrm{G}_\mathrm{F}\) gets modified by the \(\lbrace c_{\ell \ell }, c_{Hl}^{(3)} \rbrace \) operators. The oneloop renormalisation of \(\mathrm{G}_\mathrm{F}\) in SMEFT has very recently been calculated in Ref. [63].
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
In order to derive physical quantities, it is convenient to look further than the EFT Lagrangian, to Smatrix elements, since those are the gauge invariant objects that can be projected into experimental quantities. The most general EFT amplitude can be expressed as
where \(\bar{g} = g/\Lambda ^2\). The first EFT term corresponds to one \(\mathrm {dim}=6\) insertion in the original treelevel diagram, the second term represents two \(\mathrm {dim}=6\) insertions, then one insertion and one loop, two insertions and one loop, then one \(\mathrm {dim}=8\) insertion at tree level and so on, as illustrated in Fig. 1.
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.
The perturbative EFT expansion, then, grows in different directions: higherdimensional operators or higher loops. Whether the treelevel \(\mathrm {dim}=8\) correction is larger than the NLO\(\mathrm {dim}=6\) term will depend on the energy of the next new physics scale \(\Lambda \) which, up to now, is unknown. In this work, the leading order for an EFT amplitude is defined as \({\mathcal {A}}= {\mathcal {A}}_{\mathrm{SM}}+ g_6 {\mathcal {A}}_6^{(1,1)}\). Still, some ambiguities appear when squaring the amplitude in (2.8),
The term \( \sim \bar{g}^2 \vert {\mathcal {A}}_6^{(1,1)} \vert ^2 \), is commonly called “quadratic EFT” in the literature. It can be used as an estimate for the theoretical uncertainty; it is interesting to study it in depth too, since it is positive definite and hence it can have relevant effects on the differential distributions and on the unitarity bounds of the EFT expansion. This term will be studied in the context of VBS in Sect. 4.2.
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^{Footnote 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.
Regarding such cuts, there are different VBS regions that are widely accepted, but they obviously lead to different results. In the most recent LHC results, in Ref. [80], the VBS(ZZ) is defined as the purely electroweak component of \(p p \rightarrow Z Z j j \rightarrow \ell \bar{\ell } \, \ell ' \bar{\ell '} j j \), measured in the region defined by the following cuts:

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

\( m_{jj} > 100\) GeV.
One can define a VBSenriched region, with the additional cuts:

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

\( m_{jj} > 400\) GeV,
which was also used in some parts of that analysis. In this work we chose a compromise between both regions, and apply the cuts:

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

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

\( m_{jj} > 100\) GeV.
As anticipated, further cuts impose the requirement that two Z bosons be onshell, to remove the VBF contamination. Nevertheless it is important to keep in mind that the experimental cut defining “onshell” Z’s (\( M_{\ell \bar{\ell }} \in \left[ 60 , 120 \right] \,\mathrm{GeV}\) ) is not strictly the same thing as the theory definition for the Z pole.
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
To allow a straightforward comparison to be made with the existing literature, in this section we study the impact of a handful of EFT operators. In particular, we study the operators that directly affect triple and quartic gauge vertices, and that are CPeven.^{Footnote 3} We have:

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

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

\({\mathcal {O}}_\mathrm{HWB} = H^{\dagger } \tau ^I H W_{\mu \nu }^I B^{\mu \nu }\).
For this preliminary study we generated \(3 \times 10^5\) events for the process defined in Sect. 2.3 using a modified version of the SMEFTsim package [105], interfaced with Madgraph5\(_{}\)aMC@NLO [114] via FeynRules [115] and MadAnalysis5 [116]. We studied the impact of each of the three TGC/QGC operators separately, as well as the sum of them, following the definition of leadingorder EFT given in Sect. 2.1.
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.
The cases of transverse momentum distributions are very interesting themselves. We find that the EFT effects get enhanced on the tails of such distributions, which is something that we would have expected a priori for all four observables, but is not so pronounced as expected for the invariant masses. For the two cases \(p_T (Z_1)\) and \(p_T (j_1)\) we also reach the regime where the SM production is negligible but the EFT effects remain.
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).^{Footnote 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,^{Footnote 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].
In the LHC case, we find that the fit is more precise and its predictions are compatible with the ones here. The total contribution including its error bar, however, still departs from the SM expected value. Furthermore, it is important to recall that only a subset of EFT operators is included in the former fits, and hence some directions in the phase space have not been tested. The results shown on Fig. 3 further show that only one of the fourfermion operators (\({\mathcal {O}}_{\ell \ell }\), the one entering \(\mathrm{G}_\mathrm{F}\)) is well constrained, whereas other 11 of these operators enter the VBS cross section. Contrary to the common belief that fourfermion operators are maximally constrained from LEP measurements, we see that, while this might be the case for most of the leptonic operators, it is not the case for fourquark ones.
4.2 Linear and quadratic contributions
Another issue that should be handled with care is the treatment of quadratic \(\mathrm {dim}=6\) terms. This effect was discussed in Sect. 2.1, and it was thoroughly studied in Sect. I.4.7 of Ref. [47] and in Refs. [124, 125]. The quadratic contribution to the cross section is not included in the SMEFT predictions, as a matter of consistency: its perturbative order is higher than the linear term, equivalent to the \(\mathrm {dim}=8\) operators, which are not included either. However, this quadratic term is, by definition, always positive, and hence can have a large impact on the behaviour of the final distributions. There are some situations where the quadratic contribution should be carefully studied:

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, \({\mathcal {O}}(\Lambda ^4)\).
The latter situation, where \(\mathrm {dim}=8\) operators where compared with \(\mathrm {dim}=6\) quadratic terms, was recently studied in Ref. [126]. In Fig. 4 we show an example of the impact of including the quadratic contribution in some key distributions, for the canonical value \(\Lambda = 1 \,\,\mathrm{\, TeV}\). Some good news is that, for higher values of the New Physics scale, \(\Lambda \), the ratio between quadratic and linear contributions is expected to be less pronounced. That is, \(r = \frac{\left( 1/\Lambda ^2 \right) }{\left( 1/\Lambda \right) } = \frac{1}{\Lambda }\), hence \(r_2 \ll r_1\) for \(\Lambda _2 \gg \Lambda _1\).
5 The Warsaw basis in VBS
5.1 EFT for the full process
In this section, we investigate the effects of including all the Warsawbasis operators on the computation of the VBS(ZZ) cross section. Some examples of the Feynman diagrams that contribute to the process are shown in Fig. 5.
In particular we found, numerically, the following expressions for the bosonic contribution to the total cross section:
For the fermionic contribution we have
Both of this expressions have been extracted from a simple numerical analysis of relatively small Monte Carlo samples and hence are dominated by the MC uncertainty. Those we estimate to be of the order of 10% for each of the interference terms. The only purpose of displaying them here is to give an impression of the relative sensitivity of this process to the different EFT operators.
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 1 and 2 We consider all the bosonic, CPeven operators in the Warsaw basis, and, given the relative contribution of each of them to the total (linear) cross section, given by Eq. (5.1), we study two of the (infinite) possible solutions to the equation: one that gives an \({\mathcal {O}}(10\%)\) enhancement and one that gives a \({\mathcal {O}}(10\%)\) decrease (negative interference) to the SM total cross section. This is approximately the sensitivity we have to the process in LHC Run2, and the order of the EW corrections to VBS at very high energies, see Refs. [66, 127,128,129]. In Fig. 6 we see the effects of these two benchmarks for four different VBS observables, and in Table 1 we give the values used for the numerical simulation. For this part of the study we generated \(9 \times 10^5\) events for each benchmark scenario, using the tools described in Sect. 4.
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 3, 4 and 5 As in the previous section, we find the solutions that enhance (B4) or diminish (B3) the SM cross section by approximately 10%. In particular we chose the values given in Table 2. It is important to understand that these benchmarks are solutions to Eq. (5.2), which conspire to enhance or decrease the SM cross section, but the single entries of Table 2 have no physical meaning by themselves. Additionally we show the benchmark B5, which gives rise to the same total cross section as B3, but which has very different kinematics. This study can be seen in Figs. 7 and 8.
6 Dijet observables
Dijet observables (\(m_{jj}\), \(\Delta \eta _{jj}\)) are characteristic quantities to any VBS study, since the main signature of such processes is given by their jets. Dijet data in general represent a great opportunity to constrain fourquark operators at LHC, and some attempts in this direction have been presented in Refs. [131, 132]. In this section we show the rapidity distributions corresponding to the different setups that were studied. This variable seems to be a very good pointer for NP effects in VBS.
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
For this study, we generated a sample of events of the QCD component, i.e. \({\mathcal {O}}(g_s^2)\), to the \( p p \rightarrow Z Z j j \) process. The cross section for this component is much larger than the purely EW one, although this discrepancy can be minimised in the VBS fiducial region, yielding
The sensitivity to the different \(\mathrm {dim}=6\) operators can be extracted numerically in the same way as in Eqs. (5.1)–(5.2),
This background is sensitive to fewer operators than the signal, and it is sensitive to other operators \(\lbrace {\mathcal {O}}_\mathrm{HG}, {\mathcal {O}}_{\widetilde{\mathrm{HG}}} \rbrace \) to which the signal was blind. Some examples of Feynman diagrams for the \(\mathrm {dim}=6\) background can be seen in Fig. 10. In order to see the effects of the \(\mathrm {dim}=6\) in the background, we generated a sample of background events, including the EFT operators corresponding to the benchmark scenario B1 in Table 1, and leaving the new operators equal to zero.^{Footnote 6} The first interesting observation is that this benchmark, which produced a negative interference of \(20\%\) in the signal cross section, gives rise to an enhancement in the background cross section, of approximately \(\approx 1 \%\). Given the fact that the background cross section is 4 times larger than the signal one, Eq. (7.1), a few percent of events in the background can be equivalent to an \({\mathcal {O}}(20\%)\) number of signal events.
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.
In Fig. 11 we show the corresponding plots for the background study.
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^{Footnote 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.
Moreover, if we assume factorisation of the EFT effects, by defining the amplitude as in Sect. 2.1, with one insertion per diagram and at tree level, the fiducial cross section can be written as
where \(\mathcal {F}\) is a linear function of the Wilson coefficients \(\bar{c}_i\), and A is a global factor modulating the EFT contribution. In that case, the most important task is to find the relationship between the coefficients, \(\mathcal {F}\), since the global factor A can be constrained with the measurement of the fiducial cross section. The best way to do that includes using multivariate analysis (MVA) techniques, like the ones currently used in the experimental VBS searches, for example in the analysis discussed here, or in the analysis of the VBS(WZ) in ATLAS, in Ref. [135], where a 15 variable BDT was used.
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.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]
Notes
This interpretation is consistent with the fact that the EFT coefficients \(c_i\) run with the energy scale, just as every other Lagrangian parameter.
The analysis can easily be extended to the CPodd case, by adding \({\mathcal {O}}_{\widetilde{\mathrm{W}}}\), \({\mathcal {O}}_{\widetilde{\mathrm{HW}}}\) and \({\mathcal {O}}_{\widetilde{\mathrm{HWB}}}\).
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.
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.
The operator \({\mathcal {O}}_\mathrm{HG}\) is very well understood through the gluon–gluon fusion process, and \({\mathcal {O}}_{\widetilde{\mathrm{HG}}}\) is CPodd.
We focus on the ZZ channel here, but this logic applies equally well to the other VBS channels.
References
R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, M. Spira, Effective Lagrangian for a light Higgslike scalar. JHEP 07, 035 (2013). arXiv:1303.3876 [hepph]
A. Azatov, C. Grojean, A. Paul, E. Salvioni, Taming the offshell Higgs boson. Zh. Eksp. Teor. Fiz. 147, 410–425 (2015). arXiv:1406.6338 [hepph]
A. Azatov, C. Grojean, A. Paul, E. Salvioni, Taming the offshell Higgs boson. J. Exp. Theor. Phys. 120, 354 (2015)
R. Contino, M. Ghezzi, C. Grojean, M. Mühlleitner, M. Spira, eHDECAY: an implementation of the Higgs effective Lagrangian into HDECAY. Comput. Phys. Commun. 185, 3412–3423 (2014). arXiv:1403.3381 [hepph]
L. Berthier, M. Trott, Towards consistent electroweak precision data constraints in the SMEFT. JHEP 05, 024 (2015). arXiv:1502.02570 [hepph]
M. Trott, On the consistent use of constructed observables. JHEP 02, 046 (2015). arXiv:1409.7605 [hepph]
R. Alonso, E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization group evolution of the standard model dimension six operators III: gauge coupling dependence and phenomenology. JHEP 04, 159 (2014). arXiv:1312.2014 [hepph]
E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization group evolution of the standard model dimension six operators II: Yukawa dependence. JHEP 01, 035 (2014). arXiv:1310.4838 [hepph]
E.E. Jenkins, A.V. Manohar, M. Trott, Naive dimensional analysis counting of gauge theory amplitudes and anomalous dimensions. Phys. Lett. B 726, 697–702 (2013). arXiv:1309.0819 [hepph]
E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization group evolution of the standard model dimension six operators I: formalism and lambda dependence. JHEP 10, 087 (2013). arXiv:1308.2627 [hepph]
E.E. Jenkins, A.V. Manohar, M. Trott, On gauge invariance and minimal coupling. JHEP 09, 063 (2013). arXiv:1305.0017 [hepph]
P. Artoisenet et al., A framework for Higgs characterisation. JHEP 11, 043 (2013). arXiv:1306.6464 [hepph]
A. Alloul, B. Fuks, V. Sanz, Phenomenology of the Higgs effective lagrangian via Feynrules. JHEP 04, 110 (2014). arXiv:1310.5150 [hepph]
J. Ellis, V. Sanz, T. You, Complete Higgs sector constraints on dimension6 operators. JHEP 07, 036 (2014). arXiv:1404.3667 [hepph]
A. Falkowski, F. Riva, Modelindependent precision constraints on dimension6 operators. JHEP 02, 039 (2015). arXiv:1411.0669 [hepph]
I. Low, R. Rattazzi, A. Vichi, Theoretical constraints on the Higgs effective couplings. JHEP 04, 126 (2010). arXiv:0907.5413 [hepph]
C. Degrande, N. Greiner, W. Kilian, O. Mattelaer, H. Mebane, T. Stelzer, S. Willenbrock, C. Zhang, Effective field theory: a modern approach to anomalous couplings. Ann. Phys. 335, 21–32 (2013). arXiv:1205.4231 [hepph]
C.Y. Chen, S. Dawson, C. Zhang, Electroweak effective operators and Higgs physics. Phys. Rev. D 89(1), 015016 (2014). arXiv:1311.3107 [hepph]
R. Grober, M. Muhlleitner, M. Spira, J. Streicher, NLO QCD corrections to Higgs pair production including dimension6 operators. JHEP 09, 092 (2015). arXiv:1504.06577 [hepph]
C. Englert, M. McCullough, M. Spannowsky, Combining LEP and LHC to bound the Higgs width. Nucl. Phys. B 902, 440–457 (2016). arXiv:1504.02458 [hepph]
C. Englert, I. Low, M. Spannowsky, Onshell interference effects in Higgs boson final states. Phys. Rev. D 91(7), 074029 (2015). arXiv:1502.04678 [hepph]
C. Englert, A. Freitas, M.M. Mühlleitner, T. Plehn, M. Rauch, M. Spira, K. Walz, Precision measurements of Higgs couplings: implications for new physics scales. J. Phys. G41, 113001 (2014). arXiv:1403.7191 [hepph]
A. Biekötter, A. Knochel, M. Krämer, D. Liu, F. Riva, Vices and virtues of Higgs effective field theories at large energy. Phys. Rev. D 91, 055029 (2015). arXiv:1406.7320 [hepph]
R.S. Gupta, A. Pomarol, F. Riva, BSM primary effects. Phys. Rev. D 91(3), 035001 (2015). arXiv:1405.0181 [hepph]
J. EliasMiro, J.R. Espinosa, E. Masso, A. Pomarol, Higgs windows to new physics through d=6 operators: constraints and oneloop anomalous dimensions. JHEP 11, 066 (2013). arXiv:1308.1879 [hepph]
J. EliasMiro, J.R. Espinosa, E. Masso, A. Pomarol, Renormalization of dimensionsix operators relevant for the Higgs decays \(h\rightarrow \gamma \gamma ,\gamma Z\). JHEP 08, 033 (2013). arXiv:1302.5661 [hepph]
A. Pomarol, F. Riva, Towards the ultimate SM Fit to close in on Higgs physics. JHEP 01, 151 (2014). arXiv:1308.2803 [hepph]
E. Masso, An effective guide to beyond the standard model physics. JHEP 10, 128 (2014). arXiv:1406.6376 [hepph]
J. de Blas, M. Chala, M. PerezVictoria, J. Santiago, Observable effects of general new scalar particles. JHEP 04, 078 (2015). arXiv:1412.8480 [hepph]
T. Corbett, O.J.P. Eboli, D. Goncalves, J. GonzalezFraile, T. Plehn, M. Rauch, The nonlinear Higgs legacy of the LHC run I. (2015) arXiv:1511.08188 [hepph]
T. Corbett, O.J.P. Eboli, D. Goncalves, J. GonzalezFraile, T. Plehn, M. Rauch, The Higgs legacy of the LHC run I. JHEP 08, 156 (2015). arXiv:1505.05516 [hepph]
R. Contino, A. Falkowski, F. Goertz, C. Grojean, F. Riva, On the validity of the effective field theory approach to SM precision tests. JHEP 07, 144 (2016). arXiv:1604.06444 [hepph]
D. Barducci et al., Interpreting topquark LHC measurements in the standardmodel effective field theory. (2018) arXiv:1802.07237 [hepph]
M. Chala, G. Durieux, C. Grojean, L. de Lima, O. Matsedonskyi, Minimally extended SILH. JHEP 06, 088 (2017). arXiv:1703.10624 [hepph]
J. de Blas, J.C. Criado, M. PerezVictoria, J. Santiago, Effective description of general extensions of the standard model: the complete treelevel dictionary. JHEP 03, 109 (2018). arXiv:1711.10391 [hepph]
J. de Blas, O. Eberhardt, C. Krause, Current and future constraints on Higgs couplings in the nonlinear effective theory. JHEP 07, 048 (2018). arXiv:1803.00939 [hepph]
B. Henning, X. Lu, H. Murayama, How to use the standard model effective field theory. JHEP 01, 023 (2016). arXiv:1412.1837 [hepph]
M. Boggia, R. GomezAmbrosio, G. Passarino, Low energy behaviour of standard model extensions. JHEP 05, 162 (2016). arXiv:1603.03660 [hepph]
F. del Aguila, Z. Kunszt, J. Santiago, Oneloop effective lagrangians after matching. Eur. Phys. J. C 76(5), 244 (2016). arXiv:1602.00126 [hepph]
T. Appelquist, J. Carazzone, Infrared singularities and massive fields. Phys. Rev. D 11, 2856 (1975)
C. Arzt, Reduced effective Lagrangians. Phys. Lett. B 342, 189–195 (1995). arXiv:hepph/9304230 [hepph]
C. Arzt, M.B. Einhorn, J. Wudka, Patterns of deviation from the standard model. Nucl. Phys. B 433, 41–66 (1995). arXiv:hepph/9405214 [hepph]
H. Georgi, Effective field theory. Ann. Rev. Nucl. Part. Sci. 43, 209–252 (1993)
M.S. Chanowitz, M. Golden, H. Georgi, Lowenergy theorems for strongly interacting W’s and Z’s. Phys. Rev. D 36, 1490 (1987)
M.B. Einhorn, J. Wudka, The bases of effective field theories. Nucl. Phys. B 876, 556–574 (2013). arXiv:1307.0478 [hepph]
I. Brivio, M. Trott, The standard model as an effective field theory (2017). arXiv:1706.08945 [hepph]
M. Boggia et al., The HiggsTools handbook: a beginners guide to decoding the Higgs sector. J. Phys. G45(6), 065004 (2018). arXiv:1711.09875 [hepph]
A.V. Manohar, Introduction to effective field theories, in Les Houches summer school: EFT in Particle Physics and Cosmology Les Houches, Chamonix Valley, France, July 328, 2017 (2018). arXiv:1804.05863 [hepph]
P.W. Higgs, Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508–509 (1964)
F. Englert, R. Brout, Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321–323 (1964)
G.S. Guralnik, C.R. Hagen, T.W.B. Kibble, Global conservation laws and massless particles. Phys. Rev. Lett. 13, 585–587 (1964)
P.W. Higgs, Spontaneous symmetry breakdown without massless bosons. Phys. Rev. 145, 1156–1163 (1966)
T.W.B. Kibble, Symmetry breaking in nonAbelian gauge theories. Phys. Rev. 155, 1554–1561 (1967)
S. Weinberg, Baryon and Lepton nonconserving processes. Phys. Rev. Lett. 43, 1566–1570 (1979)
B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, Dimensionsix terms in the standard model lagrangian. JHEP 10, 085 (2010). arXiv:1008.4884 [hepph]
B. Henning, X. Lu, T. Melia, H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485,.: higher dimension operators in the SM EFT. JHEP 08, 016 (2017). arXiv:1512.03433 [hepph]
O.J.P. Eboli, M.C. GonzalezGarcia, J.K. Mizukoshi, p p –> j j e+ mu+ nu nu and j j e+ mu+ nu nu at O( alpha(em)**6) and O(alpha(em)**4 alpha(s)**2) for the study of the quartic electroweak gauge boson vertex at CERN LHC. Phys. Rev. D 74, 073005 (2006). arXiv:hepph/0606118 [hepph]
O.J.P. Eboli, M.C. GonzalezGarcia, Classifying the bosonic quartic couplings. Phys. Rev. D 93(9), 093013 (2016). arXiv:1604.03555 [hepph]
K. Hagiwara, S. Ishihara, R. Szalapski, D. Zeppenfeld, Lowenergy effects of new interactions in the electroweak boson sector. Phys. Rev. D 48, 2182–2203 (1993)
M. Ghezzi, R. GomezAmbrosio, G. Passarino, S. Uccirati, NLO Higgs effective field theory and \(\kappa \)framework. JHEP 07, 175 (2015). arXiv:1505.03706 [hepph]
A. Helset, M. Paraskevas, M. Trott, Gauge fixing the standard model effective field theory. Phys. Rev. Lett. 120(25), 251801 (2018). arXiv:1803.08001 [hepph]
A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek, K. Suxho, Feynman rules for the standard model effective field theory in R\(_{xi}\) gauges. JHEP 06, 143 (2017). arXiv:1704.03888 [hepph]
S. Dawson, P.P. Giardino, Higgs decays to \(ZZ\) and \(Z\gamma \) in the standard model effective field theory: an NLO analysis. Phys. Rev. D 97(9), 093003 (2018). arXiv:1801.01136 [hepph]
E. Accomando, A. Ballestrero, A. Belhouari, E. Maina, Isolating vector boson scattering at the LHC: gauge cancellations and the equivalent vector boson approximation vs complete calculations. Phys. Rev. D 74, 073010 (2006). arXiv:hepph/0608019 [hepph]
M. Rauch, VectorBoson fusion and VectorBoson scattering (2016). arXiv:1610.08420 [hepph]
B. Biedermann, A. Denner, M. Pellen, Large electroweak corrections to vectorboson scattering at the Large Hadron Collider. Phys. Rev. Lett. 118(26), 261801 (2017). arXiv:1611.02951 [hepph]
U. Baur, E.W.N. Glover, Z Boson pair production via vector boson scattering and the search for the Higgs Boson at Hadron supercolliders. Nucl. Phys. B 347, 12–66 (1990)
A. Ballestrero, D.Buarque Franzosi, L. Oggero, E. Maina, Vector Boson scattering at the LHC: counting experiments for unitarized models in a full six fermion approach. JHEP 03, 031 (2012). arXiv:1112.1171 [hepph]
R. Kleiss, W.J. Stirling, Anomalous Highenergy behavior in Boson fusion. Phys. Lett. B 182, 75 (1986)
K. Philippides, W.J. Stirling, Restoring good highenergy behavior in Higgs production via W fusion at the LHC. Eur. Phys. J. C 9, 181–195 (1999). arXiv:hepph/9901451 [hepph]
A. Ballestrero, D Buarque Franzosi, E. Maina, Exploring alternative symmetry breaking mechanisms at the LHC with 7, 8 and 10 TeV total energy. JHEP 05, 083 (2012). arXiv:1203.2771 [hepph]
W. Kilian, T. Ohl, J. Reuter, M. Sekulla, Highenergy vector boson scattering after the Higgs discovery. Phys. Rev. D 91, 096007 (2015). arXiv:1408.6207 [hepph]
R.L. Delgado, A. Dobado, F.J. LlanesEstrada, Unitarity, analyticity, dispersion relations, and resonances in strongly interacting \(W_LW_L\), \(Z_LZ_L\), and hh scattering. Phys. Rev. D 91(7), 075017 (2015). arXiv:1502.04841 [hepph]
A. Dobado, F.K. Guo, F.J. LlanesEstrada, Production cross section estimates for stronglyinteracting electroweak symmetry breaking sector resonances at particle colliders. Commun. Theor. Phys. 64, 701–709 (2015). arXiv:1508.03544 [hepph]
M. Sekulla, W. Kilian, T. Ohl, J. Reuter, Effective field theory and unitarity in vector boson scattering. PoS LHCP 2016, 052 (2016). arXiv:1610.04131 [hepph]
R. GomezAmbrosio, Study of VBF/VBS in the LHC at 13 TeV, the EFT approach, in 28th Rencontres de Blois on Particle Physics and Cosmology Blois, France, May 29June 3, 2016. (2016). arXiv:1610.07491 [hepph]
D Buarque Franzosi, Implications of vector boson scattering unitarity in composite Higgs models. PoS EPSHEP 2017, 264 (2017)
S. Brass, C. Fleper, W. Kilian, J. Reuter, M. Sekulla, Transversal modes and Higgs bosons in electroweak vectorBoson scattering at the LHC (2018). arXiv:1807.02512 [hepph]
R. L. Delgado, A. Dobado, F. J. LlanesEstrada, Unitarized HEFT for strongly interacting longitudinal electroweak gauge bosons with resonances, in 6th Large Hadron Collider Physics Conference (LHCP 2018) Bologna, Italy, June 49, 2018. (2018). arXiv:1808.04413 [hepph]
CMS Collaboration, A. M. Sirunyan et al., Measurement of vector boson scattering and constraints on anomalous quartic couplings from events with four leptons and two jets in protonproton collisions at sqrt(s) = 13 TeV, arXiv:1708.02812 [hepex]
K.J.F. Gaemers, G.J. Gounaris, Polarization amplitudes for e+ e –> W+ W and e+ e –> Z Z. Z. Phys. C 1, 259 (1979)
A. Dobado, D. Espriu, M.J. Herrero, Chiral Lagrangians as a tool to probe the symmetry breaking sector of the SM at LEP. Phys. Lett. B 255, 405–414 (1991)
B. Grinstein, M.B. Wise, Operator analysis for precision electroweak physics. Phys. Lett. B 265, 326–334 (1991)
K. Hagiwara, S. Ishihara, R. Szalapski, D. Zeppenfeld, Lowenergy constraints on electroweak three gauge boson couplings. Phys. Lett. B 283, 353–359 (1992)
J. Wudka, Electroweak effective Lagrangians. Int. J. Mod. Phys. A 9, 2301–2362 (1994). arXiv:hepph/9406205 [hepph]
M. Fabbrichesi, M. Pinamonti, A. Tonero, A. Urbano, Vector boson scattering at the LHC: a study of the WW \(\rightarrow \) WW channels with the Warsaw cut. Phys. Rev. D 93(1), 015004 (2016). arXiv:1509.06378 [hepph]
A. Falkowski, M. GonzalezAlonso, A. Greljo, D. Marzocca, M. Son, Anomalous triple gauge couplings in the effective field theory approach at the LHC. JHEP 02, 115 (2017). arXiv:1609.06312 [hepph]
T. Corbett, O.J.P. Éboli, M.C. GonzalezGarcia, Unitarity constraints on dimensionsix operators. Phys. Rev. D 91(3), 035014 (2015). arXiv:1411.5026 [hepph]
T. Corbett, M.J. Dolan, C. Englert, K. Nordström, Anomalous neutral gauge boson interactions and simplified models. Phys. Rev. D 97(11), 115040 (2018). arXiv:1710.07530 [hepph]
H. Mebane, N. Greiner, C. Zhang, S. Willenbrock, Constraints on electroweak effective operators at one loop. Phys. Rev. D 88(1), 015028 (2013). arXiv:1306.3380 [hepph]
R. Rahaman, R.K. Singh, Constraining anomalous gauge boson couplings in \(e^+e^\rightarrow W^+W^\) using polarization asymmetries with polarized beams (2017). arXiv:1711.04551 [hepph]
DELPHI, OPAL, ALEPH, L3 Collaboration, J. Jousset, Triple Couplings Of Electroweak Gauge Bosons At LEP2, in Proceedings, 33rd Rencontres de Moriond 98 electrowek interactions and unified theories: Les Arcs, France, Mar 1421, 1998, pp. 77–82, Edition Frontieres. Edition Frontieres, Paris, (1998)
DELPHI, OPAL, ALEPH, L3 Collaboration, B. de la Cruz, Production of Z pairs at LEP2 and neutral anomalous couplings, in Results and perspectives in particle physics. Proceedings, Les Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy, 2002, pp. 681–691 (2002)
ALEPH Collaboration, S. Schael et al., Improved measurement of the triple gaugeboson couplings gamma W W and Z W W in e+ e collisions, Phys. Lett. B 614, 7–26 (2005)
DELPHI Collaboration, J. Abdallah et al., Study of W boson polarisations and Triple Gauge boson Couplings in the reaction e+e —> W+W at LEP 2, Eur. Phys. J. C54, 345–364 (2008). arXiv:0801.1235 [hepex]
ALEPH Collaboration, S. Schael et al., Measurement of Zpair production in e+ e collisions and constraints on anomalous neutral gauge couplings, JHEP 04, 124 (2009)
ATLAS Collaboration, M. Aaboud et al., \(ZZ \rightarrow \ell ^{+}\ell ^{}\ell ^{\prime +}\ell ^{\prime }\) crosssection measurements and search for anomalous triple gauge couplings in 13 TeV \(pp\) collisions with the ATLAS detector, Phys. Rev. D 97(3), 032005 (2018). arXiv:1709.07703 [hepex]
ATLAS Collaboration, T. A. collaboration, \(ZZ\rightarrow \ell ^{+}\ell ^{}\ell ^{\prime +} \ell ^{\prime }\) crosssection measurements and aTGC search in 13 TeV \(pp\) collisions with the ATLAS detector,
C.M.S. Collaboration, A.M. Sirunyan et al., Measurements of the \({{\rm pp}}\rightarrow {{\rm ZZ}}\) production cross section and the \({{\rm Z}}\rightarrow 4\ell \) branching fraction, and constraints on anomalous triple gauge couplings at \(\sqrt{s} = 13\,\text{TeV} \). Eur. Phys. J. C 78, 165 (2018). arXiv:1709.08601 [hepex]. [Erratum: Eur. Phys. J. C78, no.6,515(2018)]
ALEPH Collaboration, A. Collaboration, Constraints on anomalous quartic gauge boson couplings from photon pair events from 189 GeV to 209 GeV, in Lepton and photon interactions at high energies. In: Proceedings, 20th International Symposium, LP 2001, Rome, Italy, July 2328, 2001 (2001)
O.P.A.L. Collaboration, G. Abbiendi et al., Constraints on anomalous quartic gauge boson couplings from nu antinu gamma gamma and q antiq gamma gamma events at LEP2. Phys. Rev. D 70, 032005 (2004). arXiv:hepex/0402021 [hepex]
M. Bordone, A. Greljo, G. Isidori, D. Marzocca, A. Pattori, Higgs Pseudo observables and radiative corrections. Eur. Phys. J. C 75(8), 385 (2015). arXiv:1507.02555 [hepph]
A. David, G. Passarino, Through precision straits to next standard model heights. Rev. Phys. 1, 13–28 (2016). arXiv:1510.00414 [hepph]
A. Greljo, G. Isidori, J.M. Lindert, D. Marzocca, Pseudoobservables in electroweak Higgs production. Eur. Phys. J. C 76(3), 158 (2016). arXiv:1512.06135 [hepph]
I. Brivio, Y. Jiang, M. Trott, The SMEFTsim package, theory and tools. JHEP 12, 070 (2017). arXiv:1709.06492 [hepph]
J. S. Gainer et al., Adding pseudoobservables to the fourLepton experimentalist’s toolbox (2018) arXiv:1808.00965 [hepph]
LHC Higgs Cross Section Working Group Collaboration, D. de Florian et al., Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector (2016) arXiv:1610.07922 [hepph]
C. Degrande, A basis of dimensioneight operators for anomalous neutral triple gauge boson interactions. JHEP 02, 101 (2014). arXiv:1308.6323 [hepph]
B. Bellazzini, F. Riva, \(ZZ\) and \(Z\gamma \) still haven’t found what they are looking for (2018) arXiv:1806.09640 [hepph]
J. Kalinowski, P. Kozów, S. Pokorski, J. Rosiek, M. Szleper, S. Tkaczyk, Samesign WW scattering at the LHC: can we discover BSM effects before discovering new states? Eur. Phys. J. C 78(5), 403 (2018). arXiv:1802.02366 [hepph]
C. Zhang, S.Y. Zhou, Positivity bounds on vector boson scattering at the LHC (2018) arXiv:1808.00010 [hepph]
R.L. Delgado, A. Dobado, D. Espriu, C. GarciaGarcia, M.J. Herrero, X. Marcano, J.J. SanzCillero, Production of vector resonances at the LHC via WZscattering: a unitarized EChL analysis. JHEP 11, 098 (2017). arXiv:1707.04580 [hepph]
E. Arganda, C. GarciaGarcia, M. J. Herrero, Probing the Higgs selfcoupling through double Higgs production in vector boson scattering at the LHC (2018). arXiv:1807.09736 [hepph]
J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.S. Shao, T. Stelzer, P. Torrielli, M. Zaro, The automated computation of treelevel and nexttoleading order differential cross sections, and their matching to parton shower simulations. JHEP 07, 079 (2014). arXiv:1405.0301 [hepph]
A. Alloul, N .D. Christensen, C. Degrande, C. Duhr, B. Fuks, FeynRules 2.0—a complete toolbox for treelevel phenomenology. Comput. Phys. Commun. 185, 2250–2300 (2014). arXiv:1310.1921 [hepph]
E. Conte, B. Fuks, G. Serret, MadAnalysis 5, a userfriendly framework for collider phenomenology. Comput. Phys. Commun. 184, 222–256 (2013). arXiv:1206.1599 [hepph]
L. Berthier, M. Bjørn, M. Trott, Incorporating doubly resonant \(W^\pm \) data in a global fit of SMEFT parameters to lift flat directions. JHEP 09, 157 (2016). arXiv:1606.06693 [hepph]
I. Brivio, M. Trott, Scheming in the SMEFT... and a reparameterization invariance!. JHEP 07, 148 (2017). arXiv:1701.06424 [hepph]. [Addendum: JHEP05,136(2018)]
C. Englert, R. Kogler, H. Schulz, M. Spannowsky, Higgs coupling measurements at the LHC. Eur. Phys. J. C 76(7), 393 (2016). arXiv:1511.05170 [hepph]
C. Englert, R. Kogler, H. Schulz, M. Spannowsky, Higgs characterisation in the presence of theoretical uncertainties and invisible decays. Eur. Phys. J. C 77(11), 789 (2017). arXiv:1708.06355 [hepph]
J. de Blas, M. Ciuchini, E. Franco, S. Mishima, M. Pierini, L. Reina, L. Silvestrini, The global electroweak and Higgs fits in the LHC era. PoS EPS–HEP 2017, 467 (2017). arXiv:1710.05402 [hepph]
J. Ellis, C.W. Murphy, V. Sanz, T. You, Updated global SMEFT fit to Higgs. Diboson and electroweak data. JHEP 06, 146 (2018). arXiv:1803.03252 [hepph]
S. Alioli, W. Dekens, M. Girard, E. Mereghetti, NLO QCD corrections to SMEFT dilepton and electroweak Higgs boson production, matched to parton shower in POWHEG (2018). arXiv:1804.07407 [hepph]
J. Brehmer, A. Freitas, D. LopezVal, T. Plehn, Pushing Higgs effective theory to its limits. Phys. Rev. D 93(7), 075014 (2016). arXiv:1510.03443 [hepph]
A. Biekötter, J. Brehmer, T. Plehn, Extending the limits of Higgs effective theory. Phys. Rev. D 94(5), 055032 (2016). arXiv:1602.05202 [hepph]
C. Hays, A. Martin, V. Sanz, J. Setford, On the impact of dimensioneight SMEFT operators on Higgs measurements (2018). arXiv:1808.00442 [hepph]
A. Denner, S. Pozzorini, One loop leading logarithms in electroweak radiative corrections. 1. Results. Eur. Phys. J. C 18, 461–480 (2001). arXiv:hepph/0010201 [hepph]
B. Biedermann, A. Denner, S. Dittmaier, L. Hofer, B. Jäger, Electroweak corrections to \(pp \rightarrow \mu ^+\mu ^e^+e^ + X\) at the LHC: a Higgs background study. Phys. Rev. Lett. 116(16), 161803 (2016). arXiv:1601.07787 [hepph]
B. Biedermann, A. Denner, M. Pellen, Complete NLO corrections to W\(^{+}\)W\(^{+}\) scattering and its irreducible background at the LHC. JHEP 10, 124 (2017). arXiv:1708.00268 [hepph]
D. Liu, A. Pomarol, R. Rattazzi, F. Riva, Patterns of strong coupling for LHC searches. JHEP 11, 141 (2016). arXiv:1603.03064 [hepph]
O. Domenech, A. Pomarol, J. Serra, Probing the SM with Dijets at the LHC. Phys. Rev. D 85, 074030 (2012). arXiv:1201.6510 [hepph]
S. Alte, M. König, W. Shepherd, Consistent searches for SMEFT effects in nonresonant Dijet events. JHEP 01, 094 (2018). arXiv:1711.07484 [hepph]
CMS Collaboration, A. M. Sirunyan et al., Measurement of differential cross sections for Z boson pair production in association with jets at \(\sqrt{s}=\) 8 and 13 TeV (2018). arXiv:1806.11073 [hepex]
R. GomezAmbrosio, Vector boson scattering studies in CMS: the \(pp \rightarrow ZZ jj\) channel. Acta Phys. Polon. Supp. 11, 239–248 (2018). arXiv:1807.09634 [hepph]
ATLAS Collaboration, T. A. collaboration, Observation of electroweak \(W^{\pm }Z\) boson pair production in association with two jets in pp collisions at \(\sqrt{s}\) = 13TeV with the ATLAS Detector (2018)
A. Ballestrero, E. Maina, G. Pelliccioli, \(W\) boson polarization in vector boson scattering at the LHC. JHEP 03, 170 (2018). arXiv:1710.09339 [hepph]
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}
About this article
Cite this article
GomezAmbrosio, R. Studies of dimensionsix EFT effects in vector boson scattering. Eur. Phys. J. C 79, 389 (2019). https://doi.org/10.1140/epjc/s1005201968932
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s1005201968932