Nikhef-2020-039 , IPPP/20/71

VBSCAN-PUB-01-21

SMEFT analysis of vector boson scattering
and diboson data from the LHC Run II

Jacob J. Ethier,1,2 Raquel Gomez-Ambrosio,3,4,† Giacomo Magni,1,2 Juan
Rojo,1,2

1 Department of Physics and Astronomy, Vrije Universiteit Amsterdam,
NL-1081 HV Amsterdam, The Netherlands
2 Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands
3 Dipartimento di Fisica, Universita degli Studi di Milano Bicocca
and INFN Sezione di Milano Bicocca, Milan, Italy.
4 Institute for Particle Physics Phenomenology , Durham University,
South Road DH1 3LE Durham, UK

Abstract

We present a systematic interpretation of vector boson scattering (VBS) and
diboson measurements from the LHC in the framework of the dimension-six
Standard Model Effective Field Theory (SMEFT). We consider all available
measurements of VBS fiducial cross-sections and differential distributions
from ATLAS and CMS, in most cases based on the full Run II luminosity, and use
them to constrain 16 independent directions in the dimension-six EFT parameter
space. Compared to the diboson measurements, we find that VBS provides
complementary information on several of the operators relevant for the
description of the electroweak sector. We also quantify the ultimate EFT reach
of VBS measurements via dedicated projections for the High Luminosity LHC. Our
results motivate the integration of VBS processes in future global SMEFT
interpretations of particle physics data.

† corresponding author<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Theoretical framework
  3. 3 Experimental data and theoretical calculations
    1. 3.1 Vector boson scattering
    2. 3.2 Diboson production
    3. 3.3 Sensitivity on the dimension-six EFT operators
  4. 4 Results and discussion
    1. 4.1 Fitting strategy
    2. 4.2 Fit quality and comparison with data
    3. 4.3 Constraints on the EFT parameter space
    4. 4.4 Dataset dependence
  5. 5 Vector boson scattering at the HL-LHC
  6. 6 Summary and outlook

##  1 Introduction

Since the dawn of the Standard Model (SM), the vector boson scattering (VBS)
process has been heralded as a cornerstone to test the high-energy behaviour
of the electroweak sector. Such importance originated in calculations of
scattering amplitudes involving longitudinally polarised vector bosons which,
in the absence of a Higgs boson, were shown to grow quadratically with energy
and eventually violate unitarity bounds [1, 2, 3, 4, 5, 6]. The ability to
fully scrutinise the VBS process was therefore one of the motivations to
project the ill-fated Superconducting Super Collider (SSC) with a center of
mass energy of $\sqrt{s}=40$ TeV [7]. If the Higgs boson were not responsible
for electroweak symmetry breaking, the SSC might have been able to discover
new resonances in the high-energy tail of VBS events.

While we know now that the Higgs boson, following its discovery in 2012 [8,
9], unitarises the VBS cross-sections, such processes still provide unique
sensitivity to deformations of the SM at high energies, such as those
parametrised by the Standard Model Effective Field Theory (SMEFT) [10, 11,
12]. VBS therefore provides a fully complementary probe to investigate the
electroweak sector of the SMEFT compared to processes such as on-shell Higgs
production or gauge-boson pair production, both in terms of covering a
different energy regime (up to the TeV scale) and by its contributions from
different EFT operator combinations. A particularly attractive feature of VBS
in this context is the appearance of quartic gauge couplings (QGCs), which
have often led to a theoretical interpretation of VBS data in terms of
anomalous QGCs (aQGCs).

One significant challenge in studying the VBS process at the LHC is the rather
small signal-to-noise ratios due to its electroweak nature, with backgrounds
being dominated by QCD-induced diboson production. Fortunately, VBS also
benefits from a characteristic signature that allows for a relatively clean
isolation, defined by two energetic jets in the forward region and a large
rapidity gap between them that contains reduced hadronic activity.111 This is
same kinematic signature relevant to identify single Higgs [13] and Higgs pair
[14] production in vector boson function (VBF). The combination of this
characteristic topology together with the improved analysis of the high
statistics delivered during Run II of the LHC ($\mathcal{L}=140$ fb-1 at
$\sqrt{s}=13$ TeV) has made possible not only the identification of VBS events
with reasonable statistical significance, but also the measurement of the
associated unfolded cross-sections and differential distributions in the
fiducial region [15, 16, 17, 18, 19, 20, 21, 22]. In particular, VBS
measurements from ATLAS and CMS based on the full Run II dataset have recently
been presented for different final states, from $W^{\pm}W^{\pm}jj$ and
$ZW^{\pm}jj$ [15] to $ZZjj$ [19, 18], including one analysis targeting
polarized $W^{\pm}W^{\pm}$ scattering [22].

In the past, searches for new physics using VBS processes have either been
based on unitarisation techniques [23, 24, 25, 26] or interpreted in terms of
anomalous gauge couplings, where the SM couplings are rescaled by
phenomenological parameters fitted from the data [27, 28, 29, 19]. However,
this approach is only beneficial for bookkeeping purposes since, among other
limitations, it violates gauge invariance. For this reason, different
strategies based on effective field theories have been advocated [30, 31, 32,
33] to interpret multi-boson and VBS measurements. These EFT-based approaches
have numerous advantages over the previous phenomenological approaches: they
respect the fundamental symmetries of the SM, are systematically improvable in
perturbation theory, allow the correlation of eventual deviations between
different processes, and can accommodate a meaningful quantification of
theoretical uncertainties. We note that, beyond the SMEFT, other effective
theory interpretations of VBS data have been considered such as those based on
the Electroweak Chiral Lagrangian [34, 35, 36, 37], where the Higgs boson is
not necessarily part of an SU(2) doublet.

With this motivation, VBS measurements have often been interpreted in the
SMEFT framework to identify, parametrise, and correlate possible deviations in
the structure of the electroweak gauge couplings compared to the SM
predictions. However, these studies have so far [38, 39, 40, 41] been mostly
restricted to a selection of dimension-eight operators [31, 42], in particular
those that induce aQGCs without modifying the Triple Gauge Couplings (TGCs).
As emphasized in Ref. [43], it is theoretically inconsistent to derive bounds
on aQGCs from VBS data accounting for dimension-eight operators while
neglecting the dimension-six ones, which also modifying the electroweak
interactions that enter the same observables. The fact that available EFT
interpretations of VBS processes ignore the contribution from dimension-six
operators casts doubts on the robustness of the obtained aQGCs bounds.

While several works have investigated the effects of dimension-six operators
on diboson production [44, 45, 46], including the impact of QCD corrections to
the EFT cross-sections [47, 48, 49], much less attention has been devoted to
the corresponding effects on VBS processes [50, 43, 51, 52]. In this work, we
present for the first time a systematic interpretation of VBS fiducial cross-
sections and unfolded differential distributions from the LHC in the framework
of the dimension-6 SMEFT at linear order,
$\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{-2}}\right)$, in the
effective theory expansion. Our study is carried out within the SMEFiT
framework, a toolbox for global EFT interpretations of experimental data which
has been deployed to characterise the top-quark sector [53] and is currently
being updated to perform a combined EFT analysis of Higgs boson, top-quark,
and diboson measurements from LEP and the LHC in Ref. [54].

In the present study, we consider all available VBS measurements of fiducial
cross-sections and distributions, in most cases based on the full Run II
integrated luminosity. These are complemented by the most updated QCD-induced
diboson production datasets from ATLAS and CMS [55, 56, 57, 58, 59], which are
interpreted simultaneously within the same EFT theoretical framework as the
VBS measurements. We demonstrate how the VBS measurements provide
complementary information on several operators relevant for the description
the electroweak sector of the SMEFT, in particular those modifying the triple
and quartic gauge couplings. In addition, we quantify the impact of the VBS
data by direct fits and by using statistical metrics such as information
geometry and principal component analysis. We also highlight the consistency
between the constraints separately provided by the VBS and diboson data on the
dimension-six operators considered, representing a non-trivial stress-test of
the gauge sector of the SMEFT. Overall, our analysis motivates the systematic
inclusion of VBS data in global SMEFT interpretations [60, 61, 62, 63, 64, 65,
66, 67, 68, 69].

While we have now the first VBS unfolded measurements of cross-sections and
differential distributions, they are limited by statistics. Accessing the full
physics potential associated to VBS processes will only be achieved with the
analysis of the complete dataset from the High Luminosity LHC [70, 71]. In
particular, the HL-LHC will provide access to the high energy region of
$VV^{\prime}\to VV^{\prime}$ scattering and has the potential to disentangle
contributions from $V_{L}V_{L}^{\prime}$ polarised scattering [72, 73, 74,
75]. To quantify this impact, we present projections for the reach in the EFT
parameter space of the VBS measurements expected at the HL-LHC, which
demonstrate a significant increase in sensitivity compared to current
measurements.

The structure of this paper is as follows. First, we present the theoretical
framework of the analysis in Sect. 2, in particular our definition of the
dimension-six operator basis and the flavour assumptions. In Sect. 3 we
describe the VBS and diboson data used as input for our EFT fit, outline the
details of the corresponding SM theoretical calculations, and present
different measures of the expected operator sensitivity. The main results of
this work are then presented in Sect. 4, where we derive bounds on the
relevant operators and discuss the interplay between the various data sets.
Finally, we study in Sect. 5 the impact that future measurements of VBS
processes at the HL-LHC will have on the EFT parameter space, followed by a
summary and indication of possible future developments in Sect. 6.

##  2 Theoretical framework

In this section we introduce the dimension-six SMEFT operators that will be
considered for the interpretation of the vector boson scattering and diboson
measurements at the LHC. Restricting ourselves to dimension-six operators, we
can express the SMEFT Lagrangian as,

$\mathcal{L}_{\rm SMEFT}=\mathcal{L}_{\rm SM}+\sum_{i=1}^{n_{\rm
op}}{{c_{i}}\over{\Lambda^{2}}}\mathcal{O}_{i}^{(6)}\,,$ (2.1)

where the $\mathcal{O}_{i}^{(6)}$ represent a complete basis of operators
built upon the SM fields with mass dimension equal to six, and $c_{i}$ are
their corresponding Wilson coefficients. These operators respect the
fundamental symmetries of the SM such as gauge and Lorentz invariance. In Eq.
(2.10), $\Lambda$ indicates the energy scale that determines the regime of
validity of the EFT approximation. For instance, $\Lambda$ can be interpreted
as the typical mass of the new heavy particles that arise in the ultraviolet
(UV) completion of the SM. Note that, from a bottom-up phenomenological
analysis, only the ratio $c_{i}/\Lambda^{2}$ can be determined, rather than
the two parameters separately.

In this work, we will focus on those operators that modify the interactions of
the electroweak gauge bosons. These will involve the weak gauge field strength
tensors

$W^{I}_{\mu\nu}=\partial_{\mu}W^{I}_{\nu}-\partial_{\nu}W^{I}_{\mu}-g_{2}\epsilon^{IJK}W^{J}_{\mu}W^{K}_{\nu}\,,$
(2.2) $B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}\,,$ (2.3)

as well as the SM covariant derivative, given by

$D_{\mu}=\partial_{\mu}+ig_{2}{{\sigma^{I}}\over{2}}W^{I}_{\mu}+ig_{1}Y_{f}B_{\mu}\,,$
(2.4)

where $g_{1},g_{2}$ are the weak couplings, $\sigma^{I}$ are the Pauli
matrices (SU(2)L generators), and $Y_{f}$ is the fermionic hypercharge. Here
we neglect strong interaction effects, which play a limited role in the
description of the VBS process, and set to zero the masses of all leptons and
quarks except for the top quark. Some of the relevant dimension-six operators
for this analysis will also involve the Higgs doublet field, defined in the
unitary gauge by

$\varphi={{1}\over{\sqrt{2}}}\begin{pmatrix}0\\\ v+h\end{pmatrix}\,,$ (2.5)

with $v=246$ GeV being the Higgs vacuum expectation value (vev) and $h$
represents the $m_{h}=125$ GeV Higgs boson. Here we will also consider CP-odd
operators, which are constructed in terms of the dual field strength tensors,
defined by

$\widetilde{X}_{\mu\nu}={{1}\over{2}}\epsilon_{\mu\nu\rho\sigma}X^{\rho\sigma}\,,$
(2.6)

and whose presence leads to CP-violating effects which are potentially
observable in the electroweak sector [76, 77, 78, 79, 80]

There exist several bases that span the SMEFT operator space at dimension-six.
In this work we adopt the Warsaw basis [81], which contains 59 operators for
one fermion generation, and consider only those operators that contain at
least one electroweak gauge field. This means, in particular, that we neglect
the contributions from four-fermion operators as well as from those that
modify the Yukawa interactions and the Higgs self-coupling.

#### Flavour assumptions.

In this work, we will assume that the operator structure is the same across
the three fermionic families, the so-called SU(3)5-symmetric model. In other
words, we assume flavour universality of the UV-complete theory. In practice,
this means that all Warsaw basis operators that contain fermion generation
indices will be understood as diagonal and summed over generations, e.g.,

$[c_{\varphi
f}]_{ij}(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{f}_{i}\gamma^{\mu}f_{j})\longrightarrow
c_{\varphi
f}\sum_{i=1}^{3}(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{f}_{i}\gamma^{\mu}f_{i})\,.$
(2.7)

Note that, as a consequence of this SU(3)5 symmetric flavour structure, when
comparing with constraints obtained in EFT fits based on more general flavour
specific operators, such as those that single out the top quark, the value of
our coefficient will be the average of the flavour-dependent coefficients in
that analysis.

#### Purely bosonic operators.

To begin, we define the purely bosonic operators that modify the gauge
structure of the theory as compared to the SM. In Table 2.1 we list the
dimension-six operators constructed from bosonic fields that modify the
interactions of the electroweak gauge bosons and which are considered in this
work. For each operator, we indicate its definition in terms of the SM fields
and also the notation conventions adopted both for the operator and for the
Wilson coefficient. Note that, as mentioned above, we consider both CP-even
and CP-odd operators.

The only CP-even modifications of the triple and quartic gauge couplings arise
from $\mathcal{O}_{W}$. In addition, we account for possible CP-odd
contributions to the aTGC and aQGC from the
$\\{\mathcal{O}_{\widetilde{W}},\mathcal{O}_{\varphi\widetilde{W}},\mathcal{O}_{\varphi\widetilde{B}},\mathcal{O}_{\varphi\widetilde{W}B}\\}$
operators. The remaining operators in this category modify the Higgs-gauge
($hVV$ and $hhVV$) vertices. They appear in the processes either by means of
Higgs decays (through the interference of $gg\to h\to 4\ell/2\ell 2\nu$ with
diboson production), or through the $t$-channel Higgs exchange contributions
to the VBS cross-sections. Furthermore, the operators $\mathcal{O}_{\varphi
WB}$ and $\mathcal{O}_{\varphi D}$ also enter the definitions of the gauge
masses and mixing angle in the SMEFT Lagrangian, and are hence both dependent
of our scheme choice.

CP properties | Operator | Coefficient | Definition
---|---|---|---
CP-even | $\mathcal{O}_{W}$ | $c_{W}$ | $\epsilon^{IJK}W^{I\nu}_{\mu}W^{J\rho}_{\nu}W^{K\mu}_{\rho}$
$\mathcal{O}_{\varphi W}$ | $c_{\varphi W}$ | $(\varphi^{\dagger}\varphi-{{v^{2}}\over{2}})W^{I}_{\mu\nu}W^{I\mu\nu}$
$\mathcal{O}_{\varphi B}$ | $c_{\varphi B}$ | $(\varphi^{\dagger}\varphi-{{v^{2}}\over{2}})B_{\mu\nu}B^{\mu\nu}$
$\mathcal{O}_{\varphi WB}$ | $c_{\varphi WB}$ | $\mathopen{}\mathclose{{}\left(\varphi^{\dagger}\sigma_{I}\varphi}\right)W^{I}_{\mu\nu}B^{\mu\nu}$
$\mathcal{O}_{\varphi D}$ | $c_{\varphi D}$ | $(\varphi^{\dagger}D^{\mu}\varphi)^{*}(\varphi^{\dagger}D_{\mu}\varphi)$
CP-odd | $\mathcal{O}_{\widetilde{W}}$ | $c_{\widetilde{W}}$ | $\epsilon^{IJK}\widetilde{W}^{I\nu}_{\mu}W^{J\rho}_{\nu}W^{K\mu}_{\rho}$
$\mathcal{O}_{\varphi\widetilde{W}}$ | $c_{\varphi\widetilde{W}}$ | $(\varphi^{\dagger}\varphi)\widetilde{W}^{I}_{\mu\nu}W^{I\mu\nu}$
$\mathcal{O}_{\varphi\widetilde{W}B}$ | $c_{\varphi\widetilde{W}B}$ | $\mathopen{}\mathclose{{}\left(\varphi^{\dagger}\sigma_{I}\varphi}\right)\widetilde{W}^{I}_{\mu\nu}B^{\mu\nu}$
$\mathcal{O}_{\varphi\widetilde{B}}$ | $c_{\varphi\widetilde{B}}$ | $(\varphi^{\dagger}\varphi)\widetilde{B}_{\mu\nu}B^{\mu\nu}$
Table 2.1: Dimension-six purely bosonic operators that modify the interactions
of the electroweak gauge bosons. For each operator, we indicate its definition
in terms of the SM fields, and the notational conventions that will be used
both for the operator and for the Wilson coefficient.

#### Two-fermion operators.

Another relevant class of dimension-six operators that modify the interactions
of the electroweak gauge bosons are those composed by two fermion fields and
two Higgs fields, where the gauge bosons enter via the covariant derivative.
These operators describe new contact interactions involving fermions with
gauge and Higgs bosons which are unrelated to the Yukawa couplings. They
generate corrections to the $V\ell\ell$ and $Vq\bar{q}$ vertices and can be
constrained, among other processes, from the electroweak precision observables
(EWPOs) measured by LEP [82]. They also generate contact interactions of the
form $hVf\bar{f}$ which affect specific Higgs boson production and decay
processes. The two-fermion operators that will be considered in this work are
listed in Table 2.2, and consist of seven CP-even operators containing each
two Higgs doublets, a covariant derivative, and two fermionic fields. In the
definition of these operators, we have introduced

$\overleftrightarrow{D}_{\mu}\equiv(D_{\mu}+\overleftarrow{D}_{\mu})\,,$ (2.8)

which is required to ensure that operators with fermionic neutral currents are
Hermitian. All the operators listed in Table 2.2 are CP-even.

Operator | Coefficient | Definition
---|---|---
$\mathcal{O}_{\varphi l}^{(1)}$ | $c_{\varphi l}^{(1)}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{l}_{j}\gamma^{\mu}l_{j})$
$\mathcal{O}_{\varphi l}^{(3)}$ | $c_{\varphi l}^{(3)}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}^{I}_{\mu}\varphi)(\bar{l}_{j}\sigma^{I}\gamma^{\mu}l_{j})$
$\mathcal{O}_{\varphi q}^{(1)}$ | $c_{\varphi q}^{(1)}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{q}_{j}\gamma^{\mu}q_{j})$
$\mathcal{O}_{\varphi q}^{(3)}$ | $c_{\varphi q}^{(3)}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}^{I}_{\mu}\varphi)(\bar{q}_{j}\sigma^{i}\gamma^{\mu}q_{j})$
$\mathcal{O}_{\varphi e}$ | $c_{\varphi e}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{e}_{j}\gamma^{\mu}e_{j})$
$\mathcal{O}_{\varphi u}$ | $c_{\varphi u}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{u}_{j}\gamma^{\mu}u_{j})$
$\mathcal{O}_{\varphi d}$ | $c_{\varphi d}$ | $\sum\limits_{\scriptscriptstyle j}i(\varphi^{\dagger}\overleftrightarrow{D}_{\mu}\varphi)(\bar{d}_{j}\gamma^{\mu}d_{j})$
Table 2.2: Dimension-six operators that modify the interactions of the
electroweak gauge bosons and that are composed by two fermion fields, two
Higgs fields, and one covariant derivative. The sum over the index $j$ runs
over the three SM generations, which are treated symmetrically in this study.

#### Dipole operators.

These operators involve the direct interactions between gauge bosons and
fermions, rather than the indirect ones that proceed via the covariant
derivative such as the operators listed in Table 2.2. They have a special
Lorentz structure connecting same-helicity fermions. In general, they do not
interfere with the SM, except for a few cases where the light Yukawa couplings
are taken to be nonzero. Since our analysis is restricted to the
$\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{-2}}\right)$ corrections to
the VBS and diboson cross-sections and we neglect quark masses, we do not need
to consider these operators here.

#### Parameter shifts and EWPOs.

Some of the dimension-six SMEFT operators generate a contribution to the
relevant electroweak parameters,

$m_{Z},m_{W},G_{F},\sin^{2}\theta_{W},\alpha_{\rm EW}\,,$ (2.9)

and depending on which input parameter scheme (IPS) one adopts, the
expressions for $\\{g_{1},g_{2},v\\}$, and hence for the resulting SM
Lagrangian and Feynman rules, will be different. The operators affecting these
electroweak input parameters are closely connected with the EWPOs and are thus
significantly constrained by the former. In particular, the $c_{\varphi
l}^{(3)}$ and $c_{ll}$ coefficients modify the definition of Fermi’s constant
$G_{F}$, while $c_{\varphi WB}$ and $c_{\varphi D}$ enter the $Z$ mass and
mixing angle. They can be well constrained through the measurement of the muon
lifetime and of the EW oblique parameter respectively [83]: $c_{\varphi WB}$
affects directly the value of the $S$ parameter, also known as $\rho$ [84],
whereas $c_{\varphi D}$ contributes to the $T$ parameter.

Several BSM and EFT fits of these EWPO have been performed in recent years
[85, 61, 86], and furthermore various LHC analyses tackle the extraction of
the same EWPOs from LHC data [87, 88, 89, 90, 91, 92], mostly relying on
Drell-Yan production and related processes. Here we choose not to account for
these constraints in our study, and constrain the coefficients of the
operators listed in Tables 2.1 and 2.2 solely from the VBS and diboson
measurements. In the future, once the VBS measurements are integrated in the
global EFT analysis, one will be able to constrain these electroweak parameter
shifts by including both the LEP’s EWPOs, the LHC Drell-Yan data directly [93,
94, 95, 96, 97, 98], and all other measurements (e.g. Higgs production)
sensitive to them.

#### Overview of fitted degrees of freedom.

We summarise in Table 2.3 the degrees of freedom considered in the present
work, categorised into purely bosonic and two-fermion operators. We also
indicate the notation that will be used in some of the plots and tables of the
following sections. We end up with $n_{\rm op}=16$ independent coefficients,
of which 9 are purely bosonic and 7 are two-fermion operators. Of the purely
bosonic operators, 5 are CP-even and 4 are CP-odd. Recall that we use
symmetric flavour assumptions and thus the operators involving quarks or
leptons are summed over the three SM generations.

Class |  DoF |  Notation
---|---|---
Purely bosonic |  $c_{\varphi B}$, $c_{\varphi W}$, $c_{\varphi WB}$ |  cpB, cpW, cpWB
(CP-even) |  $c_{\varphi D}$, $c_{WWW}$ |  cpD, cWWW
Purely bosonic |  $c_{\widetilde{W}}$, $c_{\varphi\widetilde{W}}$ |  cWWWtil, cpWtil
(CP-odd) |  $c_{\varphi\widetilde{B}}$, $c_{\varphi\widetilde{W}B}$ |  cpBtil, cpWBtil
two-fermion |  $c_{\varphi l}^{(1)}$, $c_{\varphi l}^{(3)}$, $c_{\varphi q}^{(1)}$ |  cpl, c3pl, cpq
(+ bosonic fields) |  $c_{\varphi q}^{(3)}$, $c_{\varphi u}$, $c_{\varphi d}$, $c_{\varphi e}$ |  c3pq, cpu, cpd, cpe
Table 2.3: Overview of the degrees of freedom considered in this analysis,
separated into purely bosonic (upper) and two-fermion (lower part) operators.
We also indicate the notation that will be used in some of the plots and
tables of the following sections. See Tables 2.1 and 2.2 for their definitions
in terms of the SM fields.

#### Amplitudes and cross-sections.

The dimension-six operators that compose the SMEFT Lagrangian Eq. (2.1) modify
a generic SM cross-section to be,

$\sigma_{\rm SMEFT}=\sigma_{\rm SM}+\sum_{i}^{n_{\rm
op}}{{c_{i}}\over{\Lambda^{2}}}\sigma^{(\rm eft)}_{i}+\sum_{i,j}^{n_{\rm
op}}{{c_{i}c_{j}}\over{\Lambda^{4}}}\widetilde{\sigma}^{(\rm eft)}_{ij}\,,$
(2.10)

where $\sigma_{\rm SM}$ indicates the SM prediction and the Wilson
coefficients are assumed to be real. The $\mathcal{O}(\Lambda^{-2})$ terms
arise from EFT operators interfering with the SM amplitude and in most cases
correspond to the dominant correction. For this reason, the cross-sections
$\sigma^{(\rm eft)}_{i}$ are usually denoted as the SMEFT linear interference
terms.

The third term in the RHS of Eq. (2.10) contains the quadratic contribution
arising from the square of the amplitudes involving dimension-six operators,
and scales as $\mathcal{O}(\Lambda^{-4})$. These quadratic terms are of the
same order of the dimension-eight operators that interfere with the SM
amplitudes and that modify the TGCs and QGCs. Given that we consider here only
dimension-six operators, the consistent inclusion of
$\mathcal{O}(\Lambda^{-4})$ corrections to VBS processes is left for future
work and we restrict ourselves to the linear approximation.

We note that linear EFT interference effects due to CP-odd operators remain
CP-odd, while squared CP-odd terms become CP-even and thus are difficult to
disentangle from their CP-even counterparts [99]. For this reason, it is
interesting to study CP-odd operators in processes for which the linear EFT
terms are dominant, such as the high energy bins of differential
distributions, or by looking at specific observables such as asymmetries.
Separating the impact of CP-even and CP-odd operators has been studied mostly
in the context of EFT analysis of the Higgs sector [100, 101, 99, 102, 78].

The SMEFT is defined to be valid for energies satisfying $E\ll\Lambda$. A
lower bound on the value of $\Lambda$ is given by the highest energy scale of
the data included in our fit, which as discussed in Sect. 3 turns out to be
around $E\simeq 3$ TeV. An upper bound on $\Lambda$ cannot be set from first
principles and requires the observation of a hypothetical heavy resonance. In
the rest of this paper, we will assume for simplicity $\Lambda=1$ TeV, with
the caveat that results for any other values of $\Lambda$ can be obtained by a
trivial re-scaling.

#### Interplay between VBS and diboson production.

Gauge boson pair production has been extensively studied as a precision probe
of the electroweak sector of the SM and its various extensions, first in the
context of precision SM electroweak tests at LEP and more recently in the EFT
framework and accounting for the corresponding LHC measurements [44, 45, 46,
47, 48, 49]. Since diboson production is a relatively clean process with large
cross-sections [103], fiducial cross-sections and differential distributions
have been measured with high precision by ATLAS and CMS.

Most of the dimension-six operators listed in Table 2.3 modify also the
theoretical calculation of diboson cross-sections, and thus it would seem that
VBS data might be redundant for EFT studies. While indeed dimension-six EFT
effects can be well constrained by diboson production at the LHC [44, 104],
here we will show that VBS measurement provide non-trivial, complementary
information for many of these operators. Furthermore, the role of VBS
measurememnts is only bound to increase as more data is accumulated, in
particular at the HL-LHC.

In VBS, only one CP-even operator in the Warsaw basis affects directly the
triple and quartic gauge couplings, with three more operators contributing
once CP-odd effects are allowed. Beyond these modifications of the TGCs and
QGCs, the VBS process is also sensitive to several other dimension-six
operators, given the large amount of vertices and topologies contributing to
the definition of the its final state. This is illustrated in Fig. 2.1, where
we show representative diagrams for EFT corrections to quartic and triple
gauge couplings as well as the the $t$-channel Higgs exchange contribution.

Figure 2.1: EFT corrections modifying the quartic (left panel) and triple
(middle panel) gauge couplings in vector-boson scattering, as well as the the
$t$-channel Higgs exchange contribution (right panel) and the $Vf\bar{f}$
interaction vertices. In this work we consider only final states where the
gauge bosons decay leptonically.

In the case of $WW$ diboson production at LEP, the process is sensitive to the
triple gauge couplings $ZWW$ and $\gamma WW$ at leading order in the EFT
expansion, and thus the corresponding EFT parametrisation will include the
modification of the TGC (through $c_{W}$). It will also modify the $e\bar{e}Z$
vertex and the corresponding IPS dependence, which could include $c_{\varphi
WB},c_{\varphi D}$ and $c_{\varphi l}^{(3)}$, and even some contact term of
the form $e\bar{e}WW$, generally not interfering with the SM. Similar
considerations apply for diboson production at hadron colliders, although now
a new feature appears, namely the interference with Higgs production in gluon
fusion followed by the $h\to VV$ decay. This correction induces a non-
negligible sensitivity to the $c_{\varphi B}$ and $c_{\varphi W}$ coefficients
in gauge boson pair production at the LHC. These features are illustrated in
Fig. 2.2.

(a)  (b)

Figure 2.2: Same as Fig. 2.1 for two representative EFT diagrams contributing
to diboson production: a pure diboson diagram (left) and another for which
diboson production interferes with the $h\to VV$ process (right).

##  3 Experimental data and theoretical calculations

In this section we describe the experimental data sets that will be used in
the present analysis as well as the corresponding theoretical predictions both
in the SM and at the EFT level. We also quantify the sensitivity that each of
the VBS and diboson data have on the coefficients associated to the dimension-
six operators introduced in Sect. 2.

###  3.1 Vector boson scattering

At hadron colliders, vector boson scattering occurs when two vector bosons are
radiated off incoming quark lines and scatter into another pair of vector
bosons, $VV^{\prime}\to VV^{\prime}$. The latter decay either leptonically or
hadronically, and thus the VBS amplitude will be proportional to $\alpha_{\rm
EW}^{6}$. Fig. 3.1 displays representative Feynman diagrams associated to
vector boson scattering at the LHC for the $ZZjj$ channel. The sensitivity to
quartic gauge couplings is a unique feature of this process, and in particular
the longitudinally polarised scattering amplitude $V_{L}V^{\prime}_{L}\to
V_{L}V_{L}^{\prime}$ provides a direct probe of the high-energy behaviour of
the theory. We emphasize again that QGCs represent only a fraction of the VBS
events, and thus a complete description of the process requires accounting for
EFT effects in all possible topologies, as discussed in Sect. 2.

(a)

(b)

(c)

(d)

(e)

Figure 3.1: Representative Feynman diagrams for vector boson scattering in the
$ZZjj$ final state (top row) and its main background, the QCD-induced diboson
production (bottom row).

The characteristic VBS topology is defined by two energetic jets with moderate
transverse momenta, $p_{T}\sim M_{V}/2$, which therefore are produced
relatively close to the beam pipe and appear predominantly in the forward
region of the detectors. The specific final-state signature that we will focus
on in this work is thus composed by four leptons (either charged or neutral)
and two jets in the forward region exhibiting a large invariant mass $m_{jj}$
and wide rapidity separation $\Delta y_{jj}$. Furthermore, being a purely
electroweak process, there is no color flow between the two incoming quark
lines. This implies that the central rapidity region between the two tagging
jets will have a reduced amount of hadronic activity, known as the “rapidity
gap”.

As highlighted by the bottom diagrams of Fig. 3.1, the vector boson scattering
process is affected by large backgrounds from QCD-induced diboson production
processes with similar topology, with amplitudes proportional instead to
$\alpha_{\rm EW}^{4}\alpha_{s}^{2}$. The interference terms between the
diboson and VBS processes are usually small and therefore will be neglected in
this analysis. Beyond diboson production, other sources of background to VBS
include $t+V$, $t\bar{t}$, $V$+jets and QCD multijet production and are
generally small. While the diboson inclusive cross-section is much larger than
the VBS one, provided the statistics are large enough, one can efficiently
disentangle the two processes by focusing on the large $m_{jj}$ and $\Delta
y_{jj}$ region (or related kinematic variables) where the VBS processes
dominates.

#### Theoretical simulations.

In order to evaluate the expected cross-sections and differential distribution
for the VBS (and the diboson) processes, we use two Monte Carlo generators,
MG5_aMC@NLO [105] and POWHEG-box [106, 107, 108], to generate NLO QCD matrix
elements. QCD corrections represent up to a $\mathcal{O}(100\%)$ effect for
diboson processes [109, 57, 58], while in VBS, a purely electroweak process,
they amount to a few percent [110, 111, 112, 113, 114, 115, 116], although
they can modify the shape of distributions. Here we adopt the NNPDF3.1NNLO no-
top PDF set [117].

The fixed-order NLO events are then showered with Pythia8 [118, 119, 120].
Accounting for parton shower effects is especially relevant for the modelling
of additional soft QCD radiation in diboson production. It is also convenient
to facilitate the matching between the theoretical predictions with the
experimental analyses. However, since we restrict ourselves to fully leptonic
final states, both hadronisation, underlying event, and multiple parton
interactions are switched off in the Pythia8 simulation. The showered events
are further processed with Rivet [121], a crucial step to reproduce the
experimental selection requirements and acceptance cuts, given that only a
subset of these can be implemented at the generation level. Moreover, this
allows us to compare directly with the datasets published in HEPData [122].

Bottom quarks are always included in the initial state ($n_{f}=5$ scheme) and
sometimes also in the definition of the final state jets, following the
prescription in the associated experimental analysis.

The signal to background ratio in VBS is generally small, and for this reason
most VBS differential results are only available as a sum of EW- and QCD-
induced processes, which can only be disentangled at the level of fiducial
cross-sections. To account for this, in the simulation of VBS processes we
generate MC events corresponding to both the EW-induced contributions (signal)
and the QCD-induced contributions (background), with EFT corrections included
only in the former222Note that EFT corrections to some of these backgrounds
are already being constrained by the diboson production measurements included
here..

The evaluation of the linear EFT cross-sections, $\sigma^{(\rm eft)}_{i}$ in
Eq. (2.10), is carried out with MG5_aMC@NLO  interfaced with SMEFTsim [123],
in its $\\{m_{W},m_{Z},G_{F}\\}$ IPS implementation. Specifically, we compute
the linear EFT cross-sections at LO in the SMEFT, and then calculate an NLO/LO
$K$-factor assuming that the QCD corrections to the SM cross-sections
factorise such that they can be assumed to be the same in the EFT.
Nevertheless, we found the impact of this assumption to be rather small at the
level of our fit results. In future work, it would be advisable to use exact
NLO QCD calculations for the EFT cross-sections, such as the ones presented in
[47, 48, 49], or by using for example SMEFT@NLO [124].

In Table 3.1 we summarize the settings of the SM and EFT theoretical
calculations used to evaluate the LHC VBS and diboson cross-sections included
in the fit. The perturbative accuracy and the codes used to produce the
corresponding predictions for both the SM and the EFT contributions are also
given.

Process | SM | Code | EFT | Code
---|---|---|---|---
$W^{\pm}W^{\mp}$ | NLO (qq), LO (gg) |  POWHEG-box [125], MG5_aMC@NLO |  LO + SM $K$-fact | SMEFTsim
$W^{\pm}Z$ | NLO |  POWHEG-box [49] |  LO + SM $K$-fact | SMEFTsim
$ZZ$ | NLO (qq), LO (gg) |  POWHEG-box [126], MG5_aMC@NLO |  LO + SM $K$-fact | SMEFTsim
$W^{\pm}W^{\pm}jj$ | LO | MG5_aMC@NLO |  LO + SM $K$-fact | SMEFTsim
$W^{\pm}Zjj$ | NLO |  POWHEG-box [127] |  LO + SM $K$-fact | SMEFTsim
$ZZjj$ | NLO |  POWHEG-box [50] |  LO + SM $K$-fact | SMEFTsim
$\gamma Zjj$ | LO | MG5_aMC@NLO |  LO + SM $K$-fact | SMEFTsim
Table 3.1: The settings of the theoretical calculations used for the
description of the LHC cross-sections included in the present analysis. We
indicate, for both the SM and the EFT contributions, the perturbative accuracy
and the codes used to produce the corresponding predictions. All the
simulations are first generated at fixed-order and then matched to a parton
shower using Pythia8.

#### Same sign $\mathbold{W^{\pm}W^{\pm}jj}$ production.

In this category we consider two data sets, one from ATLAS [16, 128] based on
$\mathcal{L}=36$ fb-1 and another from CMS based on the full Run II luminosity
[15, 129], $\mathcal{L}=137$ fb-1. Theoretical predictions are evaluated using
MG5_aMC@NLO and then showered with Pythia8. Only the fiducial cross section
measurement from ATLAS is used in the fit, since no differential distributions
are available. Concerning the CMS measurement, the input to the fit is the
differential distribution in the mass of the charged lepton pair
$m_{ll^{\prime}}$, which includes the sum of VBS (EW-induced) and diboson
(QCD-induced) contributions. In addition, we include the VBS-only fiducial
cross section measurement. To avoid double counting, we remove one bin of the
aforementioned distribution. Fig. 3.2 displays the CMS $m_{ll^{\prime}}$
measurement together with the corresponding EW+QCD-induced theoretical
predictions, finding good agreement. These theoretical predictions also agree
with those presented in the original CMS publication [15].

Figure 3.2: The dilepton invariant mass distribution $m_{ll^{\prime}}$ from
the CMS measurements of same-sign $W^{\pm}W^{\pm}jj$ production [15], compared
with the corresponding diboson (QCD-induced) plus the VBS (EW-induced)
theoretical predictions. The error bar on the CMS data points indicates the
total experimental uncertainty. In the lower panel, we display the ratio of
theory over data.

####  $\mathbold{W^{\pm}Zjj}$ production.

In this category we include the $m_{T}^{WZ}$ differential distribution from
the ATLAS measurement [17, 130] based on $\mathcal{L}=36$ fb-1, which consists
again on the sum of VBS signal and diboson background. For this dataset the
full bin-by-bin correlation matrix is available and is accounted for in the
fit. We also include the (signal plus background) differential distribution in
the dijet invariant mass from CMS, $d\sigma/dm_{jj}$, based on the full Run II
dataset luminosity of $\mathcal{L}=137$ fb-1 [15, 129]. Again, we include the
EW-only fiducial cross-section from CMS in addition to the differential
distribution, and remove a bin from the latter to avoid double counting.
Theoretical predictions for this process are evaluated at NLO with POWHEG-box
for the EW component and at LO for the QCD diboson background with
MG5_aMC@NLO.

(a)  (b)

Figure 3.3: The $W^{\pm}Zjj$ production measurements from ATLAS [17] (left)
and CMS [15] (right). In both cases the EW-induced induced contributions,
which are being added to the QCD-induced ones, are separated into $W^{+}Zjj$
and $W^{-}Zjj$.

Fig. 3.3 displays a comparison between our theoretical predictions and the
$W^{\pm}Zjj$ production measurements from ATLAS [17] (absolute $m^{WZ}_{T}$
distribution) and from CMS [15] (absolute $m_{jj}$ distribution). For
completeness, the EW-induced induced contributions, which are being added to
the QCD ones, have been separated into $W^{+}Zjj$ and $W^{-}Zjj$. In the case
of the CMS $m_{jj}$ measurement, there is good agreement between data and
theory, and one can observe how the VBS contribution clearly dominates over
the QCD-induced processes at large dijet invariant masses $m_{jj}$. For the
ATLAS measurement, we observe some tension on the second bin in $m_{T}^{WZ}$
where the theory undershoots the data, a behaviour that was also observed in
the original analysis [17]. Both the ATLAS and CMS $W^{\pm}Zjj$ measurements
benefit from sensitivity to the high-energy region, covering kinematics of up
to $m_{T}^{WZ}\simeq 1$ TeV for ATLAS and $m_{jj}=3$ TeV for CMS, which
highlights their potential for constraining EFT operators that modify the VBS
process.

####  $\mathbold{ZZjj}$ production.

Here we consider two recently released measurements from ATLAS [18, 131] and
CMS [19, 132] based on the full Run II luminosity of $\mathcal{L}\approx 140$
fb-1. The ATLAS analysis represents their first VBS measurement in the $ZZjj$
final state, while the CMS one updates a previous study of the same final
state [133]. In the ATLAS case, we include the fiducial VBS cross section,
which accounts for both EW- and QCD-induced contributions, while from CMS we
include the EW-induced fiducial cross section together with the detector-level
differential distribution in $m_{ZZ}$ for the sum of the EW and QCD-induced
contributions. Since the latter is not unfolded, it requires some modelling of
detector effects. For this reason, our baseline dataset used in the fit will
include only unfolded measurements, with the detector-level ones used as an
additional cross-check333Moreover, we only include bins of the detector-level
distribution containing more than 30 events to ensure the validity of the
Gaussian approximation..

The theoretical calculation for the $ZZjj$ process for the signal (EW-induced)
events is simulated at NLO using POWHEG-box  [50] and at LO with MG5_aMC@NLO
for the QCD-induced background. As discussed in Sect. 3.3, the $ZZjj$ final
state exhibits a large sensitivity to the EFT operators considered in this
work, but their practical impact in the fit is moderate due to the large
experimental uncertainties. In Fig. 3.4 we compare the number of events per
$m_{ZZ}$ bin between the theoretical predictions and the detector-level
experimental data from CMS in the $ZZjj$ final state based on the full Run II
luminosity. In this comparison, our simulations account for the QCD- and EW-
induced $ZZjj$ contributions, while the other sources of background are taken
from the original publication [19]. Note that the error band on the data
points includes only the statistical uncertainty, which is dominant. The
overall detector selection efficiency is modelled here by comparing the theory
prediction for the fiducial cross-section with the expected yields in the
folded distribution. In general, we observe a fair agreement between the
theory simulations and the experimental data once the experimental
uncertainties are accounted for.

Figure 3.4: The CMS detector-level VBS measurement in the $ZZjj$ final state
based on the full Run II luminosity [19]. Here we compare the number of events
per $m_{ZZ}$ bin between the theoretical predictions and the experimental
data. The error bars display only the statistical uncertainty.

####  $\mathbold{\gamma Zjj}$ production.

Finally, we consider the rare VBS final state composed by a photon $\gamma$
and a $Z$ boson which subsequently decays leptonically. In this case, we have
available two fiducial cross-section measurements for the electroweak
production of a $Z\gamma$ pair in association with two jets from ATLAS [20]
and CMS [21, 134] based on the 2016 dataset with $\mathcal{L}\simeq 36$ fb-1.
As for the $ZZjj$ final state, we will consider here one detector-level
distribution from ATLAS as a consistency check. Our theoretical predictions
for this channel are evaluated at LO with MG5_aMC@NLO and are found to be in
good agreement with the data. This channel is interesting for our study both
because of its sensitivity to neutral Higgs couplings as well as its ability
to break degenerate solutions in the EFT parameter space. Moreover, we found
that ATLAS and CMS have taken very different approaches to the definition of
the phase space, which is already useful at the level of the cross-section and
would mean an increased EFT sensitivity if unfolded distributions were also
available. In Fig. 3.5 we report the reconstructed differential distribution.
Our theoretical simulation includes only the EW signal, while other sources of
background (QCD-induced $\gamma Z$, $Z+{\rm jets}$, and $t\bar{t}\gamma$) are
taken from [20]. For this process, EW-induced VBS contributes only to $\sim
10\%$ of the total events, thus the impact of this distribution to the EFT fit
is expected to be moderate.

Figure 3.5: Comparison between data and theory predictions for the ATLAS
measurement of VBS in the $\gamma Zjj$ final state [20]. Here we compare the
number of events per $p_{T}^{\ell\ell\gamma}$ bin between our predictions and
the experimental data.

#### Overview of VBS measurements.

A summary of the VBS datasets to be considered in our EFT interpretation is
collected in Table 3.2. For each dataset, we indicate the final state, the
selection criteria (e.g. EW-only versus EW+QCD contributions), the
experimental observable, the number of data points $n_{\rm dat}$ and
integrated luminosity $\mathcal{L}$, as well as the dataset label and the
original reference. In the data labelled with (∗), one bin from the
differential distribution has been traded by the associated fiducial cross
section to avoid double counting. In those cases, the latter corresponds to
the EW-only component and thus exhibits increased sensitivity to the EFT
operators, and $n_{\rm dat}$ indicates the actual number of fitted data
points. In this overview we separate the unfolded from the folded, detector-
level data, since only the former will be part of the baseline dataset.
Overall, we end up with $n_{\rm dat}=18$ unfolded VBS cross-sections and
$n_{\rm dat}=15$ bins for the detector-level distributions, giving a total of
$n_{\rm dat}=33$ fitted data points. As will be shown in Sect. 4, the addition
of the detector-level distributions has a significant impact in a VBS-only EFT
fit, but only a marginal effect in the joint VBS+diboson analysis.

Final state | Selection | Observable | $n_{\rm dat}$ | $\mathcal{L}~{}({\rm fb}^{-1})$ | Label | Ref.
---|---|---|---|---|---|---
$W^{\pm}W^{\pm}jj$ | EW-only | $\sigma_{\rm fid}$ | 1 | 36.1 | ATLAS_WWjj_fid | [16]
EW-only | $\sigma_{\rm fid}$ | 4 | 137 | CMS_WWjj_fid | [15]
EW+QCD |  d$\sigma$/d$m_{ll}$ (∗) | CMS_WWjj_mll
$ZW^{\pm}jj$ | EW+QCD |  d$\sigma$/d$m_{T_{WZ}}$ | 5 | 36.1 | ATLAS_WZjj_mwz | [17]
EW-only | $\sigma_{\rm fid}$ | 4 | 137 | CMS_WZjj_fid | [15]
EW+QCD |  d$\sigma$/d$m_{jj}$ (∗) | CMS_WZjj_mjj
$ZZjj$ | EW+QCD | $\sigma_{\rm fid}$ | 1 | 139 | ATLAS_ZZjj_fid | [18]
EW-only | $\sigma_{\rm fid}$ | 1 | 139 | CMS_ZZjj_fid | [19]
$\gamma Zjj$ | EW-only | $\sigma_{\rm fid}$ | 1 | 36.1 | ATLAS_AZjj_fid | [20]
EW-only | $\sigma_{\rm fid}$ | 1 | 35.9 | CMS_AZjj_fid | [21]
VBS total (unfolded) |  |  | 18 |  |  |
$ZZjj$ | EW+QCD+Bkg |  Events/$m_{ZZ}$ | 4 | 139 | CMS_ZZjj_mzz | [19]
$\gamma Zjj$ | EW+QCD+Bkg |  Events/$p_{T_{\ell\ell\gamma}}$ | 11 | 36.1 | ATLAS_AZjj_ptlla | [20]
VBS total (detector-level) |  |  | 15 |  |  |
Table 3.2: Overview of the VBS measurements considered in this EFT analysis.
We indicate the final state, the selection criteria, the experimental
observable, the number of data points $n_{\rm dat}$ and integrated luminosity
$\mathcal{L}$. In the datasets labelled with (∗), one bin from the
differential distribution has been traded for the fiducial cross section. We
separate the unfolded (baseline) from the detector-level (used for cross-
checks) datasets.

###  3.2 Diboson production

In this work, gauge boson pair production is defined as the process whereby,
at leading order, two vector bosons are produced on shell and then decay. This
implies that the tree-level scattering amplitude will be proportional to
$\alpha_{\rm EW}^{4}$. Higher-order QCD corrections will lead to additional
hard radiation and thus the QCD-induced $VV^{\prime}jj$ final state becomes a
background to the VBS processes. This final state scales as $\alpha_{\rm
EW}^{4}\alpha_{s}^{2}$, and therefore in general will dominate over the EW-
induced diagrams except in regions of the phase space where the VBS topology
is enhanced.

Fig. 3.6 displays representative Feynman diagrams for opposite sign
$W^{\pm}W^{\mp}$ production, a typical example of a diboson process. One can
observe how diboson production is sensitive to the TGCs at the Born level and
that the QGCs do not enter the theoretical description of this process. The
gluon-gluon-initiated contributions are usually quite suppressed in VBS-like
analysis, since their topology does not have the characteristic forward
tagging jets. In this work, we will focus on the diboson production data with
leptonic final states, in correspondence with the VBS case.

(a)  (b)  (c)  (d)

Figure 3.6: Representative Feynman diagrams for opposite-sign $W^{\pm}W^{\mp}$
diboson production, where the first two diagrams correspond to leading order
processes while other two to gluon-initiated loop-induced contributions.

The standard experimental selection cuts for diboson processes are $p_{T}$
cuts in the leading and subleading charged leptons, leptonic rapidities being
restricted to the central region, and in the presence of $W$ bosons, a cut on
the missing transverse energy, $E_{T}^{\rm miss}\gtrsim 30$ GeV. Furthermore,
additional cuts on the transverse masses of the reconstructed leptons around
$m_{W}$ and $m_{Z}$ are required to minimise the contribution from Higgs
$s$-channel production. The resulting fiducial cross-sections are relatively
large, and already at $\mathcal{L}\simeq 36$ fb-1 they become limited by
systematic uncertainties. These large cross-sections explain why unfolded
differential cross-sections for different kinematic variables have been
available for some time already.

#### Opposite-sign $\mathbold{W^{\pm}W^{\mp}}$ production.

This channel has been measured by ATLAS based on the $\mathcal{L}=36$ fb-1
[55, 135] data in the $e\mu$ final state. Several differential distributions
are available with their corresponding bin-by-bin correlation matrices. From
CMS, we include their recent measurement [56, 136] based on the same
luminosity, where events containing two oppositely charged leptons (electrons
or muons) are selected. In our EFT analysis, we will include the same
differential distribution, $m_{\mu e}$, from both ATLAS and CMS consisting of
$n_{\rm dat}=13$ data points in each case. While the ATLAS distribution is
provided as an absolute distribution, the CMS is normalised to the fiducial
cross-section. Since the EFT total cross-section is different to the SM one,
we revert this normalisation to maximise our EFT sensitivity.

Fig. 3.7 displays a comparison between our theory predictions and the
experimental data. The measurement extends up to values of the dilepton
invariant mass of $m_{e\mu}\simeq 1.5$ TeV. Here one can observe that the
inclusion of higher-order QCD and gluon-initiated contributions is essential
to achieve a good agreement with experimental data, which turns out to be
similarly good for the two data sets. Furthermore, the effect of NLO QCD
corrections is seen to be smaller for the normalised distribution than the
absolute one, indicating that the NLO $K$-factor depends only mildly on the
value of the invariant mass $m_{e\mu}$.

(a)  (b)

Figure 3.7: The $m_{e\mu}$ differential distributions in opposite-sign
$W^{\pm}W^{\mp}$ diboson production at $\sqrt{s}=13$ TeV from ATLAS (left) and
CMS (right panel). The legend indicates the values of the $\chi^{2}$ per data
point associated to different theoretical predictions: $q\bar{q}$-initiated at
LO and NLO, and the latter plus $gg$-initiated at LO.

####  $\mathbold{W^{\pm}Z}$ production.

In this channel, we consider the ATLAS [57, 137] and CMS [58] measurements at
13 TeV based on $\mathcal{L}=36$ fb-1. In particular we chose the $e\mu\mu$
final state as a benchmark, although other combinations are available. The
ATLAS and CMS $p_{T}^{Z}$ distributions contain $n_{\rm dat}=7$ and 11 data
points and their kinematic reach is $p_{T}^{Z}\sim 1$ TeV and 300 GeV,
respectively. For the ATLAS measurement, the information on the bin-by-bin
correlated systematic uncertainties is made available and therefore are
included. Moreover, we note that an EFT interpretation in terms of a subset of
dimension-six operators has been presented in the CMS analysis of Ref. [58].

We display in Fig. 3.8 the comparison to our theoretical predictions at LO and
at NLO. The latter in particular provides an excellent description to the
experimental data. Here the effects of the NLO QCD corrections are reduced in
the normalised distributions as was the case in $W^{\pm}W^{\mp}$ production.
Finally, as we will show in Sect. 4, this channel provides the strongest
bounds on the TGC/QGC operator $\mathcal{O}_{W}$.

(a)  (b)

Figure 3.8: The $Z$ boson transverse momentum distribution, $p_{T}^{Z}$, as
measured in $W^{\pm}Z$ production from ATLAS [57] and CMS [58] at 13 TeV based
on $\mathcal{L}=36$ fb-1. Note that while ATLAS provides an absolute
distribution, the CMS one is instead normalised.

####  $\mathbold{ZZ}$ production.

For this channel, we use the recent CMS measurements based on
$\mathcal{L}=137$ fb-1 corresponding to the four-lepton final state [59],
which supersedes a previous publication based on 36 fb-1 [138, 139]. For the
theoretical predictions, the $qq\rightarrow ZZ$ and $gg\rightarrow ZZ$
contributions are simulated with POWHEG-box  at NLO and with MG5_aMC@NLO  at
LO, respectively. Fig. 3.9 displays the normalized $d\sigma/dm_{ZZ}$
distribution in the fiducial phase space from this CMS $ZZ\to 4\ell$
measurement, which contains $n_{\rm dat}=8$ data points. We find that the
agreement with the normalised distribution at LO is good, and that the
contribution from the gluon-dominated diagrams is quite small. The most
updated ATLAS analysis related to the $ZZ$ final state is the measurement of
the four-lepton invariant mass spectrum at 13 TeV based on $\mathcal{L}=36$
fb-1 [140], which receives contributions also from single-$Z$ and from Higgs
production (via $h\to ZZ^{*}$ decays) and therefore is not considered further
here.

Figure 3.9: Normalized $d\sigma/dm_{ZZ}$ distribution in the fiducial phase
space from the CMS measurement based on $\mathcal{L}=137$ fb-1.

#### Overview of diboson measurements.

The diboson measurements that will be considered in this analysis are
summarised in Table 3.3. In total we have $n_{\rm dat}=52$ diboson cross-
sections from the $W^{\pm}W^{\mp}$, $W^{\pm}Z$, and $ZZ$ channels, three times
more data points than the corresponding VBS unfolded cross-sections. In Sect.
4 we will compare the impact in the EFT parameter space between these two
families of measurements.

Final state | Selection | Observable | $n_{\rm dat}$ | $\mathcal{L}~{}({\rm fb}^{-1})$ | Label | Ref.
---|---|---|---|---|---|---
$W^{\pm}W^{\mp}$ | VV |  d$\sigma$/d$m_{e\mu}$ | 13 | 36.1 | ATLAS_WW_memu | [55]
d$\sigma$/d$m_{e\mu}$ | 13 | 35.9 | CMS_WW_memu | [56]
$W^{\pm}Z$ | VV |  d$\sigma$/d$p_{T_{Z}}$ | 7 | 36.1 | ATLAS_WZ_ptz | [57]
d$\sigma$/d$p_{T_{Z}}$ | 11 | 35.9 | CMS_WZ_ptz | [58]
$ZZ$ | VV |  d$\sigma$/d$m_{ZZ}$ | 8 | 137 | CMS_ZZ_mzz | [59]
Total diboson |  |  | 52 |  |  |
Table 3.3: Overview of the diboson measurements considered in this work.

###  3.3 Sensitivity on the dimension-six EFT operators

Quantifying the sensitivity of each VBS and diboson data set to the various
dimension-six EFT operators is an important step towards understanding the fit
results. It is also relevant to understand if there are flat directions in our
fit basis, and identify which data sets will provide the dominant constraints
in the parameter space. In the following, we summarise the dependence of each
process to the EFT operators considered and determine their relative
sensitivity by means of the Fisher information. We also apply a principal
component analysis (PCA) to identify the hierarchy of directions in the
parameter space and assess the possible presence of flat directions.

#### General discussion.

In Table 3.4 we list the contributions of the dimension-six EFT operators that
constitute our fitting basis to the various VBS and diboson processes. Overall
the complementarity between the diboson and VBS can be seen, with VBS
providing direct access to the $\mathcal{O}_{\varphi B}$ and
$\mathcal{O}_{\varphi W}$ operators (and their CP-odd counterparts) which are
essentially unconstrained from diboson-only data.

Class | Operator | CP-odd? | Diboson production | Vector boson scattering
---|---|---|---|---
$W^{\pm}W^{\mp}$ | $WZ$ | $ZZ$ | $W^{\pm}W^{\pm}jj$ | $WZjj$ | $ZZjj$ | $\gamma Zjj$
Bosonic | $\mathcal{O}_{W}$ |  | ✓ | ✓ |  | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\widetilde{W}}$ | yes | ✓ | ✓ |  | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi D}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi W}$ |  | $(\checkmark)$ |  | $(\checkmark)$ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi\widetilde{W}}$ | yes | $(\checkmark)$ |  | $(\checkmark)$ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi B}$ |  |  |  | $(\checkmark)$ |  |  | ✓ | ✓
$\mathcal{O}_{\varphi\widetilde{B}}$ | yes |  |  | $(\checkmark)$ |  |  | ✓ | ✓
$\mathcal{O}_{\varphi WB}$ |  | $(\checkmark)$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi\widetilde{W}B}$ | yes | $(\checkmark)$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
2-fermion | $\mathcal{O}^{(1)}_{\varphi\ell}$ |  |  | ✓ | ✓ |  | ✓ | ✓ | ✓
$\mathcal{O}^{(3)}_{\varphi\ell}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi e}$ |  |  | ✓ | ✓ |  | ✓ | ✓ | ✓
$\mathcal{O}^{(1)}_{\varphi q}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}^{(3)}_{\varphi q}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi u}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
$\mathcal{O}_{\varphi d}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Table 3.4: We indicate which of the $n_{\rm op}=16$ dimension-six EFT
operators considered in this analysis contributes to which of the experimental
datasets. Furthermore, with $(\checkmark)$ we denote those datasets where the
corresponding operator sensitivity arises via the gluon fusion contributions,
$gg\rightarrow h\rightarrow VV$, which is known to be a very small effect.

The operators $\mathcal{O}_{W}$ and $\mathcal{O}_{\widetilde{W}}$ modify both
the TGCs and the QGCs, and thus are not relevant for the description of the
diboson production in the $ZZ$ channel. The operators $\mathcal{O}_{\varphi
D}$ and $\mathcal{O}_{\varphi WB}$ contribute to all the diboson and VBS
channels, since they lead to modifications of the SM parameters as discussed
in Sect. 2. Given that $\mathcal{O}_{\varphi B}$ modifies only couplings
involving the Higgs boson and/or $Z$ and $\gamma$, it will be unconstrained
from the $WW$ and $WZ$ diboson channels as well as from the $WWjj$ and $WZjj$
processes. The operator $\mathcal{O}_{\varphi W}$ induces additional
modifications compared to $\mathcal{O}_{\varphi B}$, contributing to diboson
processes by means of the $hZZ$ and $hWW$ vertices.

The two-fermion interaction vertices $\gamma\bar{\psi}\psi$ and
$Z\bar{\psi}\psi$ are modified by some of the two-fermion operators,
specifically by $\mathcal{O}_{\varphi l}^{(1)}$, $\mathcal{O}_{\varphi e}$,
$\mathcal{O}_{\varphi l}^{(3)}$, $\mathcal{O}_{\varphi q}^{(3)}$,
$\mathcal{O}_{\varphi q}^{(1)}$, $\mathcal{O}_{\varphi d}$, and
$\mathcal{O}_{\varphi u}$, while the $W\bar{\psi}\psi$ vertex will be affected
by $\mathcal{O}_{\varphi l}^{(3)}$ and $\mathcal{O}_{\varphi q}^{(3)}$.
Furthermore, the $pp\to VV\to 4\ell$ and $pp\to VVjj\to 4\ell jj$ processes
provide sensitivity to two-fermion operators of the form $\varphi D\psi^{2}$
in all channels except for the ones with two $WW$ bosons. Moreover, since the
experimental phase space selection in the diboson production is designed to be
orthogonal to the Higgs production, we expect that the $WW$ and $ZZ$ channel
will be less sensitive to these operators compared to VBS. This justifies why
the contributions from $\mathcal{O}_{\varphi B}$ and $\mathcal{O}_{\varphi W}$
(and their corresponding CP-odd counterparts) are negligible in this channel.

#### The Fisher information matrix.

While certainly informative, Table 3.4 does not allow one to compare the
sensitivity brought in by different data sets on a given EFT degree of
freedom. In particular, we would like to quantify the relative impact that the
diboson and VBS observables have for each coefficient. To achieve this, it is
convenient to resort to the Fisher information matrix [63, 141] which, when
restricted to linear contributions only, is given by

$I_{ij}=\sum_{m=1}^{n_{\rm
dat}}{{\sigma^{\rm(eft)}_{m,i}\sigma^{\rm(eft)}_{m,j}}\over{\delta_{{\rm
exp},m}^{2}}}\,,\quad i,j=1,\ldots,n_{\rm op}\,,$ (3.1)

where the EFT coefficients are defined in Eq. (2.10) and where $\delta_{{\rm
exp},m}$ stands for the total experimental error associated to the $m$-th data
point. In Eq. (3.1), the sum extends over all the data points that belong to a
given data set or family of processes. While the absolute values of the
entries of the Fisher matrix $I_{ij}$ are not physically meaningful (since the
overall normalisation of the EFT operators is arbitrary), the ratios of the
diagonal entries $I_{ii}$ for the $i$-th degree of freedom between two
different groups of process is well-defined, since there the operator
normalizations cancel out.

The diagonal entries of the Fisher information matrix evaluated for each of
the degrees of freedom that form our basis are displayed in Fig. 3.10. Its
entries have been normalised such that the sum over the elements of a given
row adds up to 100. We show results both for the individual groups of
processes as well as the comparison between the overall impact of the VBS and
the diboson datasets. For those entries greater than 10%, we also indicate its
numerical value in the heat map.

Figure 3.10: The diagonal entries of the Fisher information matrix, $I_{ii}$,
evaluated for each of the coefficients that form our fitting basis. We display
results separately for each channel (left) and when clustering all VBS and
diboson datasets together (right panel). For those entries greater than 10%,
we also indicate the numerical value in the heat map.

One can observe from Fig. 3.10 that the VBS data provide the dominant
sensitivity for several of the operators considered in this analysis, in
particular for three of the CP-odd ones. In general, we find that VBS process
can provide complementary information on the EFT parameter space compared to
the diboson data. Specifically, one finds that VBS measurements provide the
dominant sensitivity (more than 50% of the Fisher information) for $c_{\varphi
B}$ and $c_{\varphi W}$ (and their CP-odd versions) as well as for
$c_{\varphi\widetilde{W}B}$. Moreover, they provide a competitive sensitivity
(defined as more than 20%) for $c_{\varphi l}^{(3)}$, $c_{\varphi d}$,
$c_{\varphi D}$ and for the triple gauge operator $c_{W}$. The latter result
illustrates how VBS measurements, while still providing less information that
diboson measurements to constrain modifications of the TGCs, do indeed provide
useful information. In the case of the triple gauge operator $c_{W}$, we also
note that the $WZ$ diboson final state dominates the sensitivity, with the
contribution from the $WW$ one being negligible. In terms of identifying which
VBS final states lead to higher relative sensitivities, we observe that $ZZjj$
provides most of the information for $c_{\varphi B}$ and
$c_{\varphi\widetilde{B}}$, $W^{\pm}W^{\mp}jj$ dominates for $c_{\varphi W}$,
and $WZjj$ leads in constraining the CP-odd operators
$c_{\varphi\widetilde{W}}$ and $c_{\varphi\widetilde{W}B}$.

#### EFT benchmark points.

Another strategy to quantify the sensitivity to the different Wilson
coefficients is to compare the size of the SM and EFT cross-sections for
representative benchmark points in the parameter space. Here we present only
representative results for these comparisons, since compatible information is
found for the complete set of final states and EFT operators. In Figs. 3.11 we
display the theoretical predictions for the VBS signal (EW-induced component
only) at $\sqrt{s}=13$ TeV. We show the differential distributions for the
$\gamma Zjj$ and $W^{\pm}W^{\pm}jj$ final states based on the selection cuts
of the corresponding ATLAS reference measurements. In each case, we compare
the SM predictions with three EFT benchmark points, in which either of the
dimensionless quantities $c_{W}v^{2}/\Lambda^{2}$, $c_{\varphi
W}v^{2}/\Lambda^{2}$, or $c_{\varphi B}v^{2}/\Lambda^{2}$ are set to 0.5, and
the rest are set to zero. In the upper panels, only the EFT prediction for
$c_{W}v^{2}/\Lambda^{2}$ is shown, to improve readability. We also display in
Fig. 3.12 the corresponding comparisons for the $m_{ZZ}$ and $m_{T}^{WZ}$
distributions in the $ZZjj$ and $W^{\pm}Zjj$ final states based on the same
selection cuts as in the associated CMS measurements.

Figure 3.11: Theoretical predictions for the VBS signal (EW-induced component
only) for different final states at $\sqrt{s}=13$ TeV. We show the dilepton
$p_{T_{\gamma\ell\ell}}$ distributions for the $\gamma Zjj$ (left) and
$m_{\ell\ell}$ for $W^{\pm}W^{\pm}jj$ (right) final states based on the
selection cuts of the corresponding ATLAS measurements. In each case, we
compare the SM predictions with three EFT benchmark points in terms of the
dimensionless quantities $\bar{c}=cv^{2}/\Lambda^{2}$. Either $\bar{c}_{W}$,
$\bar{c}_{\varphi W}$, or $\bar{c}_{\varphi B}$ are set to $0.5$ and the other
coefficients to zero. In the upper panels, only the EFT prediction with
$\bar{c}_{W}=0.5$ are shown to improve readability.

Figure 3.12: The $m_{ZZ}$ and $m_{T}^{WZ}$ distributions in the $ZZjj$ and
$W^{\pm}Zjj$ final states, based on the same selection cuts as the
corresponding CMS measurements.

From the comparisons in Figs. 3.11 and 3.12, one can observe a distinct
variation in the EFT sensitivity across the specific final state and
differential distribution being considered. In the case of the $\gamma Zjj$
and $W^{\pm}W^{\pm}jj$ final states, there is good sensitivity to $c_{W}$ but
rather less for $c_{\varphi W}$ and $c_{\varphi B}$ assuming the same value
for each coefficient. Interestingly, the sensitivity to $c_{W}$ can arise both
from the low energy region as well as from the high energy tail of the
distributions. The situation concerning $c_{W}$ is similar for the $ZZjj$ and
$W^{\pm}Zjj$ final states, with the difference being that now one becomes also
sensitive to $c_{\varphi B}$, which suppresses the cross-section compared to
the SM expectation in a manner more or less independent from the kinematics.
In the case of the $c_{\varphi W}$ coefficient, the only distribution with
comparable sensitivity to the other benchmark points is $m_{T}^{WZ}$ in the
$W^{\pm}Zjj$ final state.

#### Principal component analysis (PCA).

Lastly, we use PCA in this section to identify the combinations of Wilson
coefficients which exhibit the largest and the smallest variabilities and
determine the possible presence of flat directions. While PCA is primarily
used as a dimensionality reduction tool by removing principal components with
the lowest variance, here we use its core steps based on singular value
decomposition (SVD) only for diagnosis purposes, and the EFT fitting basis
remains the same as that defined in Sect. 2. More specifically, we utilize PCA
to identify the possible presence of flat directions, assess whether there is
a large gap in the variability between the principal components, and to
determine the matching between the physical fitting basis and the principal
components.

The starting point of the principal component analysis is the matrix $K$ of
dimensions $n_{\rm dat}\times~{}n_{\rm op}$ and (dimensionless) components
$K_{mi}=\sigma^{(\rm eft)}_{m,i}/\delta_{{\rm exp},m}$, where $\delta_{{\rm
exp},m}$ is the same total experimental error that appears in the evaluation
of the Fisher information matrix. Using singular value decomposition (SVD) we
can write $K=UWV^{\dagger}$, where $U~{}(V)$ is a $n_{\rm dat}\times n_{\rm
dat}$ ($n_{\rm op}\times n_{\rm op}$) unitary matrix and $W$ is an $n_{\rm
dat}\times n_{\rm op}$ diagonal matrix with semi-positive real entries, called
the singular values, which are ordered by decreasing magnitude. The larger a
singular value, the higher the variability of the associated principal
component. The elements $V$ contain the (normalised) principal components
associated to each of the singular values, which can be expressed as a
superposition of the original coefficients,

${\rm PC}_{k}=\sum_{i=1}^{n_{\rm op}}a_{ki}c_{i}\,,\quad k=1,\ldots,n_{\rm
op}\,,\qquad\mathopen{}\mathclose{{}\left(~{}\sum_{i=1}^{n_{\rm
op}}a_{ki}^{2}=1\,~{}\forall k}\right)$ (3.2)

where the larger the value of the coefficient $a_{kl}$, the larger the
relative weight of the associated Wilson in this specific principal component.

Figure 3.13: Results of the principal component analysis applied to the
baseline VBS+diboson dataset. The upper panel shows the distribution of
singular values, while the lower one displays the squared values for the
coefficients $a_{ki}$ of the principal components in Eq. (3.2)

The upper panel of Fig. 3.13 displays the distribution of singular values for
the $n_{\rm op}=16$ principal components associated to the fitting basis
described in Sect. 2 with the baseline VBS+diboson dataset. This analysis
confirms that there are no flat directions in our parameter space, which would
appear as a principal component with a vanishing singular value. Furthermore,
we do not observe large hierarchies in the distribution of singular values,
indicating that the physical dimensionality of our problem coincides with that
of the adopted fitting basis.

The lower panel of Fig. 3.13 displays a heat map indicating the values of the
(squared) coefficients $a_{ki}^{2}$ that relate the original fitting basis to
the principal components via the rotation in Eq. (3.2), and whose associated
eigenvalues are displayed in the upper panel. For entries with $a^{2}_{ki}\geq
0.1$, we also indicate the numerical value in the corresponding entry. The
principal component associated with the highest singular value can be
attributed to the two-fermion coefficient $c_{\varphi q}^{(3)}$, which
therefore is expected to be well constrained from the fit (as anticipated in
Sect. 2). Other principal components which coincide with the coefficients of
our fitting basis are $c_{W}$ and $c_{\varphi\widetilde{B}}$. In general, the
majority of principal components involve a superposition of several basis
coefficients $c_{i}$, for example in PCk with $k=2,7,8$ or 10, none of the
squared coefficients $a_{ki}^{2}$ is larger than 0.3.

##  4 Results and discussion

In this section, we present the main results of this work, namely the
dimension-six EFT interpretation of the VBS and diboson datasets from the LHC
Run II. We first briefly summarise the fitting strategy adopted in this
analysis and then present the fit quality by comparing the best-fit results
with the corresponding experimental measurements. We then present the fit
results for the baseline dataset, determine the 95% CL intervals for the
$n_{\rm op}=16$ operators considered, and study the dependence of our results
with respect to variations of the input data, in particular with fits based
only on VBS measurements.

###  4.1 Fitting strategy

The EFT analyses carried out in this work are based on the SMEFiT global
fitting framework presented in [54, 53]. Two options to constrain the EFT
parameter are available in this framework: the Monte Carlo replica fit method
(MCfit) and Nested Sampling (NS) via MultiNest [142]. In this work we adopt
the latter technique. The end result of SMEFiT is a representation of the
probability density in the space of Wilson coefficients spanned by $N_{\rm
spl}$ samples, $\\{c_{i}^{(k)}\\}$, which allows the evaluation of statistical
estimators such as mean values and standard deviations, e.g.,

$\mathopen{}\mathclose{{}\left\langle c_{i}}\right\rangle={{1}\over{N_{\rm
rep}}}\sum_{k=1}^{N_{\rm rep}}c_{i}^{(k)}\,,\quad i=1,\ldots,n_{\rm op}\,,$
(4.1) $\delta c_{i}=\mathopen{}\mathclose{{}\left({{1}\over{N_{\rm
rep}-1}}\sum_{k=1}^{N_{\rm
rep}}\mathopen{}\mathclose{{}\left(c_{i}^{(k)}-\mathopen{}\mathclose{{}\left\langle
c_{i}}\right\rangle}\right)^{2}}\right)^{1/2}\,,\quad i=1,\ldots,n_{\rm
op}\,,$ (4.2)

and likewise for other estimators such as the correlation coefficients. Since
the present analysis is carried out at the linear level in the EFT expansion,
and there are no flat directions for the baseline dataset (see Sect. 3.3), the
probability distributions associated to the coefficients $\bm{c}$ are expected
to be Gaussian. For this reason, it is not necessary to go beyond the first
two moments of the posterior distributions in $\bm{c}$.

The overall fit quality is assessed by means of the $\chi^{2}$ figure of
merit, defined as

$\chi^{2}\mathopen{}\mathclose{{}\left({\bm{c}}}\right)\equiv{{1}\over{n_{\rm
dat}}}\sum_{i,j=1}^{n_{\rm dat}}\mathopen{}\mathclose{{}\left(\sigma^{(\rm
th)}_{i}\mathopen{}\mathclose{{}\left({\bm{c}}}\right)-\sigma^{(\rm
exp)}_{i}}\right)({\rm
cov}^{-1})_{ij}\mathopen{}\mathclose{{}\left(\sigma^{(\rm
th)}_{j}\mathopen{}\mathclose{{}\left({\bm{c}}}\right)-\sigma^{(\rm
exp)}_{j}}\right)\;,$ (4.3)

where $\sigma_{i}^{\rm(exp)}$ corresponds to the central experimental data
point and $\sigma^{(\rm
th)}_{i}\mathopen{}\mathclose{{}\left({\bm{c}}}\right)$ is the associated
theoretical prediction, Eq. (2.10), for the $i-$th cross-section. The
covariance matrix, ${\rm cov}$, is constructed from all available sources of
uncorrelated and correlated experimental uncertainties, with the ‘$t_{0}$’
definition [143] used for the fit and the standard experimental covariance
used to quote the resulting $\chi^{2}$ values. Whenever appropriate, we also
add to the covariance matrix estimates of theoretical uncertainties coming
from the input proton PDFs, as well as the MC theory calculations. The post-
fit $\chi^{2}$ values are then evaluated using the best-fit estimate (mean) of
the Wilson coefficients, Eq. (4.1), computed from the resulting MC samples
obtained by NS.

###  4.2 Fit quality and comparison with data

In Table 4.1 we display the values of the $\chi^{2}/n_{\rm dat}$, Eq. (4.3),
for each of the data sets contained in our baseline fit, as well as the total
values associated to the diboson and VBS categories. We also indicate the
$\chi^{2}$ values corresponding to the Standard Model predictions (pre-fit)
together with the values obtained once the EFT corrections are accounted for
(post-fit). Note that our baseline dataset does not contain any detector-level
folded distributions. The graphical representation of these $\chi^{2}$ values
is also displayed in Fig. 4.1.

Process |  Dataset | $\quad n_{\rm dat}\quad$ |  $\chi^{2}/n_{\rm dat}$ (SM) |  $\chi^{2}/n_{\rm dat}$ (EFT)
---|---|---|---|---
Diboson | ATLAS_WW_memu | 13 | 0.70 | 0.66
CMS_WW_memu | 13 | 1.28 | 1.32
ATLAS_WZ_ptz | 7 | 1.38 | 0.93
CMS_WZ_ptz | 11 | 1.48 | 1.14
CMS_ZZ_mzz | 8 | 1.17 | 0.74
Total diboson | 52 | 1.17 | 0.97
VBS | ATLAS_WWjj_fid | 1 | 0.01 | 0.67
CMS_WWjj_fid | 1 | 2.17 | 0.15
CMS_WWjj_mll | 3 | 0.31 | 0.45
ATLAS_WZjj_mwz | 5 | 1.60 | 1.52
CMS_WZjj_fid | 1 | 0.38 | 0.79
CMS_WZjj_mjj | 3 | 1.10 | 0.73
ATLAS_ZZjj_fid | 1 | 0.09 | 0.15
CMS_ZZjj_fid | 1 | 0.02 | 0.02
ATLAS_AZjj_fid | 1 | 0.00 | 0.25
CMS_AZjj_fid | 1 | 0.03 | 0.38
Total VBS | 18 | 0.83 | 0.75
| Total | 70 | 1.084 | 0.917
Table 4.1: The values of the $\chi^{2}/n_{\rm dat}$ for each dataset
considered in the fit, as well as the totals in each category. We indicate the
SM (pre-fit) results as well as the best-fit results once EFT effects are
accounted for, and separate the diboson (upper) from the VBS (bottom)
datasets. Figure 4.1:  Graphical representation of the $\chi^{2}$ values
reported in Table 4.1.

From Table 4.1 one can observe that for the diboson data, a $\chi^{2}$ of
around one per data point is obtained. Moreover, the total $\chi^{2}/n_{\rm
dat}=1.17$ found at the level of SM calculations is reduced to 0.97 once EFT
effects are included in the fit. Concerning the VBS dataset, there is a higher
spread in the $\chi^{2}/n_{\rm dat}$ values, which is explained by the fact
that each data set is composed of either a single or a few cross-section
measurements. Taking into account the 18 independent cross-section
measurements that we includein the fit, the SM value of $\chi^{2}/n_{\rm
dat}=0.83$ is reduced to 0.75 at the post-fit level. Overall, the combination
of the diboson and VBS measurements adds up to $n_{\rm dat}=70$ data points
for which a pre-fit value of $\chi^{2}/n_{\rm dat}=1.08$ based on the SM
predictions is reduced to 0.92 after the EFT fit.

Fig. 4.2 displays a comparison between experimental data and best-fit EFT
theory predictions for the LHC diboson distributions considered in the present
analysis. We show the results for the $W^{\pm}Z$, $W^{\pm}W^{\mp}$ and $ZZ$
final states from CMS in the upper panels and the corresponding $W^{\pm}Z$ and
$W^{\pm}W^{\mp}$ distributions from ATLAS in the lower panels. Both the data
and the EFT fit results are normalised to the central value of the SM
prediction. The experimental data is presented as both unshifted in central
values (where the error band represents the total error) and with the best-fit
systematic shifts having been subtracted (so that the error band contains only
the statistical component). The band in the EFT prediction indicates the post-
fit 95% CL uncertainty. For the datasets in which the information on
correlated systematics is not available, only the unshifted data is shown. In
Fig. 4.3, we show a similar comparison as that of Fig. 4.2 but now for the VBS
measurements. In all cases a fair agreement is observed between experimental
data and SM and EFT theory predictions, consistent with the $\chi^{2}$ values
reported in Table 4.1.

Figure 4.2: Comparison between experimental data and best-fit EFT theory
predictions for the LHC diboson distributions considered in the present
analysis. Both the data and the EFT fit results are normalised to the central
value of the SM prediction. The band in the EFT prediction indicates the post-
fit 95% CL uncertainty.

Figure 4.3: Comparison for the VBS measurements, both for the unfolded
differential distributions (top panel), and for the EW-only fiducial cross-
sections (bottom panel).

###  4.3 Constraints on the EFT parameter space

We now present the constraints on the coefficients of the dimension-six EFT
operators used to interpret the VBS and diboson cross-sections listed in Table
4.1. In Fig. 4.4, we display the posterior probability distributions
associated to each of the $16$ coefficients that are constrained in this
analysis for the baseline dataset. In all cases, we can see that these are
approximately Gaussian, as expected for a linear EFT fit without flat
directions. The latter result is consistent with observations derived from the
PCA in Fig. 3.13, and confirm that the input dataset is sufficient to
constrain all 16 independent directions in the EFT parameter space.

Figure 4.4: The posterior probability distributions associated to each of the $n_{\rm op}=16$ coefficients that are constrained in this analysis for the baseline dataset. Note that the $x$-axis ranges are different for each coefficient.  Class | Coefficient |  VBS+diboson |  VBS+diboson |  Diboson-only
---|---|---|---|---
(marginalised) |  (individual) |  (marginalised)
purely bosonic | $\bar{c}_{W}$ |  [-0.13, 0.14] |  [-0.001, 0.13] |  [-0.20, 0.11]
(CP-even) | $\bar{c}_{\varphi W}$ |  [-0.55, 1.4] |  [-0.048, 0.91] |  [-0.97, 2.1]
$\bar{c}_{\varphi B}$ |  [-11, 8.8] |  [-0.62, 0.69] |  —
$\bar{c}_{\varphi WB}$ |  [-0.13, 0.44] |  [-0.050, 0.071] |  [-0.20, 0.44]
$\bar{c}_{\varphi D}$ |  [-0.93, 0.32] |  [-0.21, 0.08] |  [-1.09, 0.26]
purely bosonic | $\bar{c}_{\widetilde{W}}$ |  [-0.35, 0.57] |  [-0.008, 0.46] |  [-0.63, 0.85]
(CP-odd) | $\bar{c}_{\varphi\widetilde{W}}$ |  [-2.9, 1.8] |  [-0.49, 0.93] |  [-4.9, 1.3]
$\bar{c}_{\varphi\widetilde{W}B}$ |  [-1.4, 1.8] |  [-0.49, 0.69] |  [-1.3, 2.4]
$\bar{c}_{\varphi\widetilde{B}}$ |  [-19, 23] |  [-1.2, 1.4] |  —
two-fermion | $\bar{c}_{\varphi l}^{(1)}$ |  [-0.56, 0.45] |  [-0.015, 0.031] |  [-1.3, 0.12]
$\bar{c}_{\varphi l}^{(3)}$ |  [-0.037, 0.051] |  [-0.024, -0.002] |  [-0.068, 0.081]
$\bar{c}_{\varphi q}^{(1)}$ |  [0.043, 0.50] |  [-0.007, 0.12] |  [0.038, 0.68]
$\bar{c}_{\varphi q}^{(3)}$ |  [-0.002, 0.011] |  [-0.006, 0.014] |  [-0.008, 0.013]
$\bar{c}_{\varphi e}$ |  [-0.58, 0.77] |  [-0.038, 0.021] |  [-1.5, 0.41]
$\bar{c}_{\varphi u}$ |  [-0.49, 0.53] |  [-0.073, 0.42] |  [-0.59, 0.58]
$\bar{c}_{\varphi d}$ |  [-1.3, 1.0] |  [-0.53, 0.39] |  [-1.4, 1.2]
Table 4.2: The 95% confidence level intervals associated to the 16 dimension-
six EFT degrees of freedom considered in the present analysis. We compare the
results of fits based on the baseline VBS+diboson dataset both at the global
(marginalised) and the individual levels, as well as with those of a fit based
only on the diboson cross-sections. Results shown here correspond to the
dimensionless quantities $\bar{c}=c\cdot v^{2}/\Lambda^{2}$ and can be
rescaled for any value of $\Lambda$, or presented as dimensioful quantities by
extracting the powers of the vev.

From these posterior probability distributions, the 95% confidence level
intervals associated to each of the fit coefficients can be evaluated. Table
4.2 displays these 95% CL intervals associated to all 16 degrees of freedom.
Moreover, a comparison is made between the results of the baseline VBS+diboson
fit performed at the global (marginalised) and individual levels, as well as
with a fit based only on the diboson cross-sections. In the fourth column
(individual fits), only one coefficient is varied at a time while all others
are set to their SM values. The results of Table 4.2 are also graphically
represented in Fig. 4.5, which displays the absolute value (upper) and the
magnitude (bottom panel) of these 95 $\%$ CL intervals.

Figure 4.5: Graphical representation of the results of Table 4.2, displaying
the absolute value (upper) and the magnitude (bottom panel) of the 95 $\%$ CL
intervals associated to each of the 16 EFT operators considered here. We
compare the marginalised results of a diboson-only fit (blue) with the same
fit once VBS data is added (orange) in both cases when all coefficients are
fitted simultaneously. For reference, we also show the results of the
individual VBS+diboson fits, where only one operators is varied at the time
and the rest are fixed to their SM value.

From the comparison between the 95% CL intervals in Table 4.2 and Fig. 4.5,
several interesting observations can be made. First, in comparing the results
of the combined VBS+diboson fit with the diboson-only analysis, the VBS
measurements are seen to improve the bounds provided by the diboson data in a
pattern consistent with the Fisher information matrix displayed in Fig. 3.10.
For instance, the bounds on $\bar{c}_{\varphi W}$ improve from
$\mathopen{}\mathclose{{}\left[-0.97,+2.1}\right]$ to
$\mathopen{}\mathclose{{}\left[-0.55,+1.4}\right]$, while those on the CP-even
(odd) triple gauge operator $\bar{c}_{W}$ ($\bar{c}_{\widetilde{W}}$) are
reduced from $\mathopen{}\mathclose{{}\left[-0.20,+0.11}\right]$
($\mathopen{}\mathclose{{}\left[-0.63,+0.85}\right]$) down to
$\mathopen{}\mathclose{{}\left[-0.13,+0.14}\right]$
($\mathopen{}\mathclose{{}\left[-0.35,+0.57}\right]$). In all cases, the VBS
data improve the bounds on the EFT coefficients obtained from the diboson-only
fit, highlighting the consistency and complementarity between the two families
of processes. This result applies both to the CP-even as well as the CP-odd
operators.

Another relevant observation from Table 4.2 concerns the differences between
the marginalised and individual fits in the case of the combined VBS+diboson
analysis, which illustrates the role of the correlations between the operators
that modify these two processes. In the individual fits, one finds more
stringent bounds by artificially setting all other EFT operators to zero, and
this distorts the physical interpretation of the results. For several
operators, the individual bounds underestimate the results of the
16-dimensional fit by an order of magnitude or more. This highlights the
importance of accounting for all relevant EFT operators that contribute to a
given process rather than just selecting a subset of them, as has often been
the case in the interpretation of VBS measurements.

Fig. 4.6 then displays the values of the correlation coefficient between the
operators considered in the fit to the baseline dataset. For some pair-wise
combination of operators we observe strong (anti-)correlations between the fit
coefficients, for example $c_{\varphi B}$ and $c_{\varphi\widetilde{B}}$ are
strongly anticorrelated, and the same holds for $c_{\varphi D}$ and
$c_{\varphi WB}$. However, in most cases, these correlations turn out to be
quite small, confirming that our choice of fitting basis is suitable to
describe efficiently the available dataset in consistency with the PCA
results.

Figure 4.6: The values of the correlation coefficients between the operators
considered in the fit to the baseline dataset. These are categorised as
positively correlated ($\rho\geq 0.50\ (0.75)$, in red (orange)), negatively
correlated ($\rho\leq 0.50\ (0.75)$, in green (blue)), and uncorrelated
($|\rho|\leq 0.5$, in grey).

Finally, in Fig. 4.7 we display the 95% CL lower bounds on the value of
$\Lambda/(v\,\sqrt{c_{i}})$. These bounds can be interpreted as the lower
bounds derived from the EFT fit on the scale of new physics $\Lambda$ in UV-
completions where the corresponding Wilson coefficients are
$c_{i}=\mathcal{O}\mathopen{}\mathclose{{}\left(1}\right)$. They are again
presented as dimensionless quantities, measured in units of vev. This
interpretation can be adjusted to other BSM scenarios, for example in the case
of strongly coupled theories where one expects
$c_{i}=\mathcal{O}\mathopen{}\mathclose{{}\left(4\pi}\right)$. For several
operators, the combined VBS+diboson analysis results in values above 1 TeV for
the new physics scale $\Lambda$, for example the triple gauge operator $c_{W}$
which has $(\Lambda/\sqrt{c_{i}})\gtrsim 3v$ at the 95% CL.

Figure 4.7: Same as the lower panel of Fig. 4.5 now representing the 95% CL
bounds on $\Lambda/(v\sqrt{c_{i}})$.

#### Comparison with other EFT analyses.

Fig. 4.8 displays a comparison between the individual bounds obtained in this
work, based on the VBS+diboson dataset and shown in Fig. 4.5, with the
corresponding individual bounds obtained in the BDHLL20 [48] and EMMSY20 [68]
EFT analyses. The BDHLL20 fit includes data on diboson cross-sections from the
LHC together with information from the associated production of a Higgs with a
vector boson, $hW$ and $hZ$. EMMSY20 is instead a global EFT interpretation
that includes Higgs and top production data together with the EWPOs from LEP
and some diboson cross-sections. For the three sets of results shown in Fig.
4.8, only the linear terms in the EFT expansion are being included and the EFT
cross-sections are evaluated at leading order.444We note that the BDHLL20
analysis has been performed also accounting for NLO QCD corrections in the EFT
cross-sections, here we use the LO ones for the sake of comparison. Given that
these three analyses are based on different subsets of dimension-six
operators, a comparison at the level of individual constraints is the most
direct way of interpreting similarities or differences. We also note that CP-
odd operators are only considered in this analysis.

For the majority of operators, the global study of EMMSY20 exhibits the
superior sensitivity. Our good determination of $c_{W}$ can be traced back to
the inclusion of the $WZ$ differential distributions from ATLAS and CMS, which
are also included in BDHLL20, but absent in EMMSY20, where $Zjj$ is included
instead. This fact hints that a combined analysis of $WZ$ and $Zjj$ might shed
more light on the purely gauge operator. The results of the global EFT fit
lead to more stringent bounds as compared to those from this work and from
BDHLL20, especially for the purely bosonic operators $c_{\varphi B}$,
$c_{\varphi W}$ and $c_{\varphi BW}$, which are significantly constrained both
by the EWPOs from LEP as well as Higgs measurements. For most coefficients,
our individual results and those of BDHLL20 are in good agreement, in
particular for bosonic operators $c_{\varphi D}$, $c_{\varphi BW}$, and
$c_{\varphi W}$, $c_{W}$. This is what we would expect, given the datasets
chosen.

The comparison of the three works shows that Higgs, LEP and EWPD measurements
represent the leading contributions to the parametrisation of BSM effects.
There are also enough hints that a global interpretation of the LHC data,
independent of older measurements is also a feasible way to go further on the
road to the most accurate EFT interpretation.

Figure 4.8: Comparison of the individual bounds obtained in this work from the
VBS+diboson dataset (shown in Fig. 4.5) with the corresponding individual
bounds obtained in the BDHLL20 [48] and EMMSY20 [68] and EFT analyses, see
text. In the three cases, only the linear terms in the EFT expansion are being
included and the EFT cross-sections are evaluated at leading order.

###  4.4 Dataset dependence

Until now, we have focused only on the analysis of the EFT fit results for the
baseline dataset listed in Table 4.1. In the following, we assess the
dependence of these results with respect to variations in the input data and
theory settings by performing VBS-only fits and studying the impact of the VBS
detector-level distributions when added to the VBS-only and to the baseline
VBS+diboson fits. We also present fits where the CP-odd operators are set to
zero and only the CP-even ones remain.

#### VBS-only fits.

First of all, we have verified through a dedicated PCA that flat directions in
the EFT parameter space are absent also in the case of a VBS-only fit .
However, the same analysis also reveals that some combinations of coefficients
will be poorly constrained. The latter result is not unexpected, given that
for a VBS-only dataset we have $n_{\rm op}=16$ parameters to fit with only
$n_{\rm dat}=18$ data points. We display in Fig. 4.9 the same 95% CL intervals
as in the lower panel of Fig. 4.5, but now comparing the results of our
baseline fit with those obtained from the marginalised and individual VBS-only
fits. By comparing the VBS+diboson with the VBS-only fits, we see that the
obtained bounds in the latter case are much looser by a factor between 10 and
100 for most operators. These findings are consistent with our previous
observations that current VBS data provides only a moderate pull when added
together with the diboson cross-sections.

Figure 4.9: Comparison of the 95% CL intervals in the baseline fit with those
resulting from marginalised and individual VBS-only fits. Only the unfolded
VBS cross-section measurements listed in Table 3.2 are being included in the
fits.

However, we would like to emphasize that this result does not imply that VBS-
only fits cannot provide competitive sensitivity in a EFT analysis, but rather
that the available VBS measurements are still scarce and limited by
statistics. In fact, if one compares the results of the marginalised with the
individual VBS-only fits, one can see that the individual bounds are notably
reduced and become similar, or even better, than in the baseline VBS+diboson
analysis. This implies that VBS processes are endowed with a unique potential
to constrain the dimension-six operators of the SMEFT, but only once
sufficient data has been collected to pin down the effects of the individual
operators separately. We will verify this expectation in Sect. 5 through EFT
fits based on dedicated HL-LHC projections.

#### The impact of the VBS detector-level measurements.

As was discussed in Sect. 3, one can in principle use detector-level
measurements in the EFT fit in addition to the unfolded VBS cross-sections and
distributions measured by ATLAS and CMS. Here we consider the $m_{ZZ}$ and
$p_{T}^{\ell\ell\gamma}$ distributions from CMS and ATLAS in the $ZZjj$ and
$\gamma Zjj$ final states respectively, which consist of 15 data points that
can be included together with the unfolded VBS cross-section measurements.
Given that our modelling of the detector response is basically reduced to a
flat acceptance correction, we have chosen to remove these data points from
the baseline results presented in the previous section. We would therefore
like to illustrate how these detector-level distributions contain valuable
information and are particularly instrumental to realise a reliable VBS-only
EFT dimension-six analysis.

Figure 4.10: Posterior distributions associated to the VBS-only fits that
include only unfolded cross-sections (blue) and also the detector-level
distributions (orange).

Fig. 4.10 displays the same posterior probability distributions as in Fig. 4.4
but now corresponding to the VBS-only fits. We compare the results of the
analysis based only on unfolded cross-sections with that in which the two
detector-level distributions mentioned above are also included. While the VBS-
only fit based on unfolded cross-sections does not exhibit genuine flat
directions, several coefficients end up poorly constrained. The situation is
different once the detector-level distributions are added to the fit: here the
posterior distributions become Gaussian-like, and their width is markedly
reduced compared to the previous case. In particular, the inclusion of the
$m_{ZZ}$ and $p_{T}^{\ell\ell\gamma}$ detector-level distributions is
particularly helpful in strengthening the VBS-only bounds on $c_{\varphi B}$
and its CP-odd counterpart.

The 95% CL intervals associated to the posterior probability distributions of
Fig. 4.10 are then represented in Fig. 4.11, where for reference we also
display the results of the baseline VBS+diboson fit. We find that by adding
the detector-level distributions, there is a noticeable improvement in the
result of the VBS-only fit, with bounds being reduced by a factor between two
and ten depending on the specific operator. In the case of $c_{\varphi B}$,
the resulting bound becomes comparable to that obtained in the VBS+diboson
fit, though in general the VBS-only fit cannot compete with the combined
VBS+diboson results even after the addition of the folded data. These results
motivate the release of all available VBS measurements in terms of unfolded
distributions. We have verified that in the case of the combined VBS+diboson
fit, adding the detector-level measurements leaves the results essentially
unaffected, providing a further justification of our choice of removing them
from the baseline dataset.

Figure 4.11: Comparison of the results of the VBS-only fit based only on
unfolded cross-sections with those of the same fit where in addition one
includes the detector-level distributions. For reference, we also display the
results for baseline VBS+diboson dataset.

#### The impact of CP-odd operators.

Finally, we assess how the EFT fit results are modified once only CP-
conserving operators are considered. Fig. 4.12 compares the results of the
baseline VBS+diboson fit with those of the same fit where the CP-odd operators
have been set to zero, such that only the CP-even ones remain. In general the
differences are quite small, and as expected the fit without CP-violating
operators leads to somewhat more stringent bounds. The only operator for which
removing the CP-odd operators has a significant effect is $c_{\varphi B}$,
where a difference of an order of magnitude in the 95% CL bound is observed.
The reason for this behaviour is that, as indicated in the correlation heat
map of Fig. 4.6, $c_{\varphi B}$ and $c_{\varphi\widetilde{B}}$ are strongly
anti-correlated and thus in general it is rather challenging to disentangle
them.

Figure 4.12: Comparing the results of the baseline fit with those of the same
fit where the CP-odd operators have been set to zero, such that only the CP-
even ones remain.

##  5 Vector boson scattering at the HL-LHC

While the results presented in the previous section indicate the potential of
VBS measurements for dimension-6 EFT analyses, their impact is currently
limited by statistics. The ultimate LHC sensitivity required to constrain the
coefficients of these dimension-6 operators from VBS data will only be
achieved by legacy measurements based on the full HL-LHC luminosity of
$\mathcal{L}\simeq 3$ ab-1 per experiment. With this motivation, we generate
HL-LHC pseudo-data for EW-induced vector boson scattering processes and
quantify their impact on the EFT fit by comparing the results to those
presented in Sect. 4.3. The strategy adopted here is the same as the one used
for the HL-LHC PDF projections in Refs. [144, 145], which were subsequently
used in the studies presented in the corresponding Yellow Reports [70, 71].

In order to generate the HL-LHC pseudo-data, we select reference measurements
out of the VBS datasets presented in Sect. 3. Table 5.1 presents the overview
of the HL-LHC projections considered in this analysis, which include only EW-
induced VBS processes since we assume that the QCD-induced backgrounds can be
removed at the analysis level. We consider the following differential
distributions for each final state: $m_{\ell\ell}$ for $W^{\pm}W^{\pm}jj$,
$p_{T}^{\ell\ell\ell}$ and $m_{T}^{WZ}$ in $ZW^{\pm}jj$, $m_{ZZ}$ for $ZZjj$,
and then $p_{T}^{\gamma\ell\ell}$ and $m_{\gamma Z}$ in the $\gamma Zjj$ final
state, yielding a total of $n_{\rm dat}=61$ datapoints. The theoretical
predictions for these observables are generated as in Sect. 3 with the same
selection and acceptance cuts, except that they are rescaled to account for
the increase in the center of mass energy from $\sqrt{s}=13$ TeV to
$\sqrt{s}=14$ TeV. We note that the actual HL-LHC analysis are expected to
contain a larger number of bins, as well as a higher reach in energy, however
for simplicity we maintain here the current binning. The theoretical
calculations are generated for the null hypothesis ($\bm{c}=\bm{0}$), with the
caveat that better sensitivities would be obtained in the case of an EFT
signal.

The statistical and systematic uncertainties associated to the HL-LHC pseudo-
data are evaluated as follows. First, we denote $\sigma^{\rm th}_{i}$ as the
theoretical prediction for the EW-induced VBS cross-section in the $i$-th bin
of a given differential distribution. This cross-section includes all relevant
selection and acceptance cuts, as well as the leptonic branching fractions.
The expected number of events in this bin and the associated (relative)
statistical uncertainty $\delta_{i}^{\rm stat}$ are then given by,

$N_{i}^{\rm th}=\sigma^{\rm th}_{i}\times\mathcal{L}\,,\quad\delta_{i}^{\rm
stat}\equiv{{\delta N_{i}^{\rm stat}}\over{N_{i}^{\rm
th}}}={{1}\over{\sqrt{N_{i}^{\rm th}}}}\,.$ (5.1)

Note that the relative statistical uncertainty for the number of events and
for the cross-sections will be the same, either in the fiducial region or
extrapolated to the full phase space. Here we take the luminosity to be
$\mathcal{L}=3$ ab-1 and generate two differential distributions per final
state, one from ATLAS and the other from CMS, as indicated in Table 5.1.

Final state | Selection | Observable | $n_{\rm dat}$ |  $\mathcal{L}$ (ab-1) | Label
---|---|---|---|---|---
$W^{\pm}W^{\pm}jj$ | EW-induced |  d$\sigma$/d$m_{ll}$ | 7 | 3 | ATLAS_WWjj_mll_HL
EW-induced |  d$\sigma$/d$m_{ll}$ | 4 | 3 | CMS_WWjj_mll_HL
$ZW^{\pm}jj$ | EW-induced |  d$\sigma$/d$p_{T_{\ell\ell\ell}}$ | 5 | 3 | ATLAS_WZjj_plll_HL
EW-induced |  d$\sigma$/d$m_{T}^{WZ}$ | 5 | 3 | CMS_WZjj_mwz_HL
$ZZjj$ | EW-induced |  d$\sigma$/d$m_{ZZ}$ | 9 | 3 | ATLAS_ZZjj_mzz_HL
EW-induced |  d$\sigma$/d$m_{ZZ}$ | 9 | 3 | CMS_ZZjj_mzz_HL
$\gamma Zjj$ | EW-induced |  d$\sigma$/d$p_{T}^{\gamma\ell\ell}$ | 13 | 3 | ATLAS_AZjj_ptlla_HL
EW-induced |  d$\sigma$/d$m_{\gamma Z}$ | 9 | 3 | CMS_AZjj_maz_HL
HL-LHC VBS total |  |  | 61 |  |
Table 5.1: Overview of the (EW-induced) VBS HL-LHC projections considered in
this analysis.

Concerning the systematic uncertainties, these are also taken from the
reference measurements as follows. If $\delta_{i,j}^{\rm sys}$ denotes the
$j$-th relative systematic uncertainty associated to the $i$-th bin of the
reference measurement, we assume that the same systematic error at the HL-LHC
will be given by $f_{{\rm red},j}\delta_{i,j}^{\rm sys}$, where $f_{{\rm
red},j}\simeq 1/2$ is the expected reduction in systematic errors, in
agreement with available projections [72, 73, 74, 75]. Adding in quadrature
all systematic uncertainties with the statistical error, the total relative
uncertainty for the $i$-th bin of our HL-LHC projections will be given by

$\delta_{{\rm tot},i}^{\rm
exp}=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\delta_{i}^{\rm
stat}}\right)^{2}+\sum_{j=1}^{n_{\rm
sys}}\mathopen{}\mathclose{{}\left(f_{{\rm red},j}\delta_{i,j}^{\rm
sys}}\right)^{2}}\right)^{1/2}\,,$ (5.2)

where $n_{\rm sys}$ indicates the number of systematic error sources. Finally,
we generate the central values for the HL-LHC pseudo-data projections by
fluctuating the theory prediction by the expected total experimental
uncertainty, namely

$\sigma^{\rm hllhc}_{i}\equiv\sigma^{\rm
th}_{i}\mathopen{}\mathclose{{}\left(1+r_{i}\delta_{{\rm tot},i}^{\rm
exp}}\right)\,,\qquad i=1,\ldots,n_{\rm bin}\,,$ (5.3)

where $r_{i}$ are univariate Gaussian random numbers. By construction, one
expects that the EFT fit quality to the HL-LHC pseudo-data to be
$\chi^{2}/n_{\rm bin}\simeq 1$ for a sufficiently large number of bins.

(a)  (b)

Figure 5.1: Comparison of the 95% CL intervals for the EFT coefficients
between three related analyses: the VBS-only and a the combined diboson+VBS
fits based on current data, and the VBS-only fit based on the HL-LHC
projections listed in Table 5.1.

Fig. 5.1 displays the comparison of the obtained 95% CL intervals for the 16
EFT coefficients considered here between three related analyses. In
particular, EFT fits based on the current measurements, both for a VBS-only
and for a combined diboson+VBS dataset, are compared with the corresponding
results from the VBS-only fit based on the HL-LHC projections listed in Table
5.1. Here we find that the HL-LHC measurements lead to a significant impact at
the level of the VBS-only fit, where the current best bounds are improved by
up to three orders of magnitude depending on the specific coefficient. It is
also interesting to note that a VBS-only fit from HL-LHC measurements would
even have a superior sensitivity compared to the combined diboson+VBS
analysis, especially for the purely bosonic operators where at least a factor
of 10 improvement over the current bounds is expected.

The results presented here further highlight the capability of VBS
measurements for dimension-six EFT studies and the relevance of their
integration in the global EFT fit, especially as more luminosity is
accumulated. While our projections are based on optimistic assumptions such as
a clean separation between the EW- and QCD-induced components of the
measurement, the outstanding performance of the LHC experiments so far is
rather encouraging.

##  6 Summary and outlook

In this work, we have presented an exhaustive investigation of effects from
dimension-six SMEFT operators in the theoretical modelling of vector boson
scattering processes. By exploiting information provided by the most updated
VBS measurements from ATLAS and CMS, several of which are based on the full
Run II data, we have obtained bounds on the relevant SMEFT operators that
contribute to this process. We have demonstrated the overall consistency of
the constraints provided by VBS with those from diboson production, and have
highlighted how VBS measurements provide a useful addition to global EFT
interpretations of LHC data. Using tailored projections, we have also
estimated the improvements in the bounds on these dimension-six operators that
can be expected from the VBS process with the legacy measurements of the HL-
LHC, finding that these measurements will provide a remarkable sensitivity to
several directions in the EFT parameter space.

We emphasize that the goal of this work was not to achieve state-of-the-art
bounds on all the dimension-six operators that modify VBS observables. Such
ambition can only be achieved within a dedicated global EFT fit that includes
all relevant sensitive observables. These analyses must include, among others,
Higgs production and decay measurements from the LHC and electroweak precision
observables from electron-positron colliders, which by virtue of the
electroweak gauge symmetry, constrain several of the same dimension-six
operators that enter the description of VBS observables, as well as Drell-Yan
distributions. For such an effort, some improvements in the theory
calculations compared to this work will be required, in particular the use of
exact, rather than approximate, NLO QCD effects in the EFT cross-sections
using SMEFT@NLO as well as accounting for the quadratic corrections in the EFT
expansion.

Most of the previous EFT interpretations of VBS observables from the LHC have
focused on dimension-eight operators, with the argument that these can modify
the quartic gauge couplings while leaving unaffected the triple ones that are
purportedly well constrained by other processes. It would therefore be
important to revisit these studies within a consistent EFT analysis that
includes the effects of both dimension-six and dimension-eight operators up to
$\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{-4}}\right)$. For instance,
it would be important to quantify how the current bounds on dimension-8
operators are modified with the inclusion of the dimension-six ones. Since
there is no cross-talk between the dim-6 and dim-8 operators at this order in
the EFT expansion, it would be possible to extend the present analysis by
adding the various sources of quadratic contributions separately. Such a fully
consistent $\mathcal{O}\mathopen{}\mathclose{{}\left(\Lambda^{-4}}\right)$
analysis, combined with future measurements from Run III and the HL-LHC, would
unlock the ultimate potential of EFT interpretations of VBS data and represent
one of the key legacy results from the LHC.

Additional avenues for future research include the EFT interpretation of novel
VBS observables, such as polarised scattering, as well as going beyond the
SMEFT by considering other effective theories such as the HEFT or the
Electroweak Chiral Lagrangian. In this respect, we point out that the fitting
framework used in this work can be straightforwardly extended to other EFTs,
and a fully general dependence of the theory predictions with the EFT
coefficients is allowed.

The first measurements of unfolded VBS cross-sections and differential
distributions discussed in this work undoubtedly represent a milestone in the
LHC program, with profound implications for our understanding of the gauge
sector in the SM and its extensions. While current VBS measurements are still
statistics-dominated and, for the time being, provide only a moderate pull in
the EFT fit, we have demonstrated that they provide complementary information
as compared to the more traditional diboson processes. VBS is therefore poised
to play a growing role in global EFT interpretations in the coming years,
especially once high-statistics measurements become available.

Acknowledgments

We are grateful to Simone Alioli, Fabio Maltoni, Giampiero Passarino, Eleni
Vryonidou, and Cen Zhang for discussions about the topics covered in this
paper and for feedback on this manuscript. We thank our colleagues of the
VBScan COST action for many useful discussions about VBS over the past years.
Special thanks go to the HEPData team, in particular Graeme Watt, and to the
ATLAS and CMS analyzers that have provided assistance with the implementation
of the VBS measurements: Claude Charlot, Roberto Covarelli, Guillelmo Gómez-
Ceballos, and Kristin Lohwasser. The work of J. E., G. M., and J. R. is
partially supported by the Netherlands Organization for Scientific Research
(NWO). R. G. acknowledges funding from the ERC Starting Grant REINVENT-714788
and from the Fondazione Cariplo and Regione Lombardia, grant 2017-2070, as
well as the UK Science and Technology Facilities Council (STFC) grant
ST/P001246/1.

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