aainstitutetext: Theoretical Particle Physics and Cosmology Group, Department
of Physics,
King’s College London, London WC2R 2LS, UKbbinstitutetext: Theoretical Physics
Department, CERN, CH-1211 Geneva 23, Switzerlandccinstitutetext: National
Institute of Chemical Physics & Biophysics, Rävala 10, 10143 Tallinn,
Estoniaddinstitutetext: DAMTP, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, UKeeinstitutetext: Instituto de Física Corpuscular (IFIC),
Universidad de Valencia-CSIC, E-46980 Valencia, Spainffinstitutetext:
Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
UKgginstitutetext: Cavendish Laboratory, University of Cambridge, J.J. Thomson
Avenue, Cambridge CB3 0HE, UK

# Top, Higgs, Diboson and Electroweak Fit to the Standard Model Effective
Field Theory

John Ellis d Maeve Madigan a Ken Mimasu e,f Veronica Sanz b,d,g and Tevong You
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The search for effective field theory deformations of the Standard Model (SM)
is a major goal of particle physics that can benefit from a global approach in
the framework of the Standard Model Effective Field Theory (SMEFT). For the
first time, we include LHC data on top production and differential
distributions together with Higgs production and decay rates and Simplified
Template Cross-Section (STXS) measurements in a global fit, as well as
precision electroweak and diboson measurements from LEP and the LHC, in a
global analysis with SMEFT operators of dimension 6 included linearly. We
present the constraints on the coefficients of these operators, both
individually and when marginalised, in flavour-universal and top-specific
scenarios, studying the interplay of these datasets and the correlations they
induce in the SMEFT. We then explore the constraints that our linear SMEFT
analysis imposes on specific ultra-violet completions of the Standard Model,
including those with single additional fields and low-mass stop squarks. We
also present a model-independent search for deformations of the SM that
contribute to between two and five SMEFT operator coefficients. In no case do
we find any significant evidence for physics beyond the SM. Our underlying
$\tt{Fitmaker}$ public code provides a framework for future generalisations of
our analysis, including a quadratic treatment of dimension-6 operators.

KCL-PH-TH/2020-73, CERN-TH-2020-202

##  1 Introduction

Experimental tests of the Standard Model (SM) and probes of possible new
physics beyond it have been taken to a new level by LHC measurements during
its Runs 1 and 2. Previous to the LHC, the most precise tests of the SM were
those provided by measurements at LEP, notably at the $Z$ peak ALEPH:2005ab
and in $W^{+}W^{-}$ diboson production Heister:2004wr ; Achard:2004zw ;
Abbiendi:2007rs ; Schael:2013ita , and at the Tevatron, notably in
measurements of the $W$ Aaltonen:2013iut  and top quark Aaltonen:2017efp . The
LHC has added new classes of precise measurements, notably those of Higgs
production and decays Khachatryan:2016vau ; Aad:2015gba ; Aad:2019mbh ;
CMS:2020gsy ; CMS:1900lgv ; CMS:2019pyn ; CMS:2019chr ; CMS:2019kqw ;
Aad:2020plj ; Aad:2020xfq  and the production of the top quark in various
modes at Run 1 Aaboud:2016iot ; Aad:2015mbv ; Khachatryan:2015oqa ;
Sirunyan:2017lvd ; Sirunyan:2017azo ; Aad:2020jvx ; Aaboud:2016hsq ;
Khachatryan:2016fky ; Aad:2015eua ; Khachatryan:2015sha ; Aaboud:2017pdi ;
Khachatryan:2016ewo ; CMS:2014ika ; Khachatryan:2014iya ; Aad:2015upn ;
Chatrchyan:2014tua ; Aad:2020zhd ; Aad:2015eto  and Run 2 Sirunyan:2017mzl ;
Sirunyan:2018wem ; ATLAS:2019czt ; Aaboud:2019njj ; Sirunyan:2017uzs ;
CMS:2019too ; Sirunyan:2019hqb ; Sirunyan:2016cdg ; Aaboud:2016ymp ;
CMS:2016xnv ; Aaboud:2016lpj ; Sirunyan:2017nbr ; Sirunyan:2018lcp ;
Aaboud:2017ylb ; Sirunyan:2018zgs ; ATLAS:2020hrf ; Aaboud:2018jsj ;
Sirunyan:2019wxt ; Sirunyan:2019nxl ; Sirunyan:2019jud ; Sirunyan:2020kga ,
the $W$ mass Aaboud:2017svj , and also triple-gauge coupling measurements in
diboson production Aaboud:2019nkz ; Sirunyan:2019gkh  and $Zjj$ production
Aad:2020sle . The SM is a tight theoretical framework that connects all these
measurements. For this reason, it is necessary to take a global approach to
the interpretation of its tests and to searches for deviations from its
predictions that may be signatures of possible new physics beyond the SM
(BSM).

The interactions within the SM are largely constrained by its
SU(3)$\times$SU(2)$\times$U(1) gauge invariance, the exceptions being the
magnitudes of the Yukawa couplings of fermions to the Higgs boson, and its
self-interactions. Currently, all experimental data are consistent with the
SU(3)$\times$SU(2)$\times$U(1) assignments of particles that are specified in
the SM. It is therefore natural to assume that any BSM interactions must be
invariant under SU(3)$\times$SU(2)$\times$U(1) and respect these established
representation assignments. The absence of any evidence for additional
particles at the LHC or elsewhere suggests that any BSM particles may be
significantly heavier than the electroweak scale, as represented by the masses
of the $W,Z$, top quark and Higgs boson. In this case, the low-energy effects
of the more massive BSM particles may be approximated by integrating them out
to obtain higher-dimensional interactions between the SM fields
Appelquist:1974tg . In this approach the SM is regarded as an effective theory
whose known renormalizable interactions are supplemented by higher-order terms
scaled by inverse powers of the BSM mass scale(s) Weinberg:1979sa ;
Buchmuller:1985jz . This SM Effective Field Theory (SMEFT) is a powerful tool
for analyzing the consistency of the SM and searching indirectly for possible
BSM physics 111A more general EFT parametrisation in which the electroweak
SU(2)$\times$U(1) symmetry is non-linearly realised and the Higgs is added as
a singlet scalar, known as the Higgs EFT (HEFT), is described, e.g., in Ref.
Cohen:2020xca  with fits given in Refs. Buchalla:2015qju ; Brivio:2016fzo ;
deBlas:2018tjm . Its geometric interpretation is developed in Refs.
Alonso:2015fsp ; Helset:2020yio ..

There have been intensive theoretical efforts in recent years to formulate
consistently the SMEFT and to classify the sets of independent operators
appearing at each dimension Weinberg:1979sa ; Buchmuller:1985jz ;
Grzadkowski:2010es ; Lehman:2014jma ; Lehman:2015coa ; Henning:2015alf ;
Gripaios:2018zrz ; Criado:2019ugp ; Murphy:2020rsh ; Li:2020gnx ; Li:2020xlh ;
Liao:2020jmn . The first non-redundant basis of dimension-6 operators, known
as the Warsaw basis, was laid out in Ref. Grzadkowski:2010es , and the matrix
of one-loop anomalous dimensions of these operators has been computed in Refs.
Jenkins:2013zja ; Jenkins:2013wua ; Alonso:2013hga . These and other
theoretical developments have laid the groundwork for phenomenological SMEFT
analyses of available data to go beyond fitting subsets of specific operators
towards global fits in a complete basis of operators, beginning with early
global SMEFT fits to electroweak precision data Han:2004az  that started to
include triple-gauge coupling measurements and the Higgs boson Pomarol:2013zra
; Corbett:2012ja ; Ellis:2014jta  soon after its discovery Aad:2012tfa ;
Chatrchyan:2012ufa . Other LHC fits focused on triple-gauge couplings and/or
Higgs measurements for subsets of operators Dumont:2013wma ; Corbett:2013pja ;
Chang:2013cia ; Elias-Miro:2013mua ; Boos:2013mqa ; Ellis:2014dva . Since then
here have been a variety of studies of the SMEFT, for example, in electroweak
processes Falkowski:2014tna ; Berthier:2016tkq ; Banerjee:2019twi ;
Biekotter:2020flu ; Banerjee:2018bio ; Falkowski:2020znk , flavour physics
Efrati:2015eaa ; Aebischer:2018iyb ; Aoude:2020dwv ; Aebischer:2020dsw ;
Aebischer:2020lsx ; Hurth:2019ula ; Descotes-Genon:2018foz ; Faroughy:2020ina
, low-energy precision data Falkowski:2015krw ; Falkowski:2017pss ;
Falkowski:2020pma , diboson measurements Grojean:2018dqj ; Gomez-
Ambrosio:2018pnl ; Panico:2017frx ; Kozow:2019txg ; Dawson:2018pyl ;
Baglio:2018bkm ; Baglio:2019uty ; Azatov:2019xxn , at dimension 8
Ellis:2017edi ; Ellis:2019zex ; Alioli:2020kez ; Li:2020gnx ; Murphy:2020rsh ;
Ellis:2020ljj ; Bellazzini:2018paj  (where collider positivity constraints are
particularly relevant Bellazzini:2018paj ; Bellazzini:2017bkb ; Fuks:2020ujk ;
Bonnefoy:2020yee ; Remmen:2019cyz ), and its connection with UV-complete
models, both at tree-level Giudice:2007fh ; deBlas:2017xtg ; Dawson:2020oco ;
Anisha:2020ggj ; Marzocca:2020jze ; Dawson:2017vgm ; Mecaj:2020opd  and one-
loop Henning:2014gca ; Drozd:2015kva ; Drozd:2015rsp ; Ellis:2020ivx ;
Haisch:2020ahr ; Chala:2020vqp ; Gherardi:2020det ; Bakshi:2018ics ;
Huo:2015nka ; Huo:2015exa ; Chiang:2015ura ; Han:2017cfr . In particular, the
most recent global analyses have set constraints on dimension-6 SMEFT operator
coefficients imposed by precision electroweak data from LEP and the Tevatron,
together with Higgs and diboson data from the LHC including some from Run 2
Ellis:2018gqa ; Almeida:2018cld ; Biekotter:2018rhp ; Falkowski:2019hvp
222See Refs. ATLAS:2020naq ; CMS:2020gsy  for recent SMEFT interpretations of
the Higgs by ATLAS and CMS., while separate SMEFT fits of data on the top
quark have also been performed Buckley:2015lku ; Brivio:2019ius ;
Bissmann:2019gfc ; Hartland:2019bjb ; Durieux:2019rbz ; vanBeek:2019evb ;
CMS:2020pnn .

Figure 1:  Schematic representation of the datasets and their overlapping
dependences on the 34 Wilson coefficients included in our analysis.

We present here the first global dimension-6 SMEFT analysis to include top
data and operators in a simultaneous combination of the constraints from the
Higgs, electroweak, diboson and top sectors. We use a full set of data from
LHC Run 2, in particular the latest Higgs Simplified Template Cross Section
(STXS) measurements, differential distributions in $WW$ diboson and $Zjj$
measurements, and updated top observables including kinematic distributions,
$t\bar{t}$, single-top and $t\bar{t}W/Z$ production. In addition to expanding
our dataset, improvements over previous fits include a proper computation
using SMEFT@NLO Degrande:2020evl  of the dimension-6 contributions to Higgs
gluon fusion in STXS bins and incorporating the full SMEFT dependence in off-
shell Higgs to 4 lepton decays Brivio:2019myy . We also provide a self-
consistent treatment of the triple-gluon operator at linear order that had
been omitted from our previous fit Ellis:2014jta  on the basis of strong
constraints at quadratic order Krauss:2016ely ; Hirschi:2018etq ;
Goldouzian:2020wdq .

Fig. 1 provides a schematic representation of the dependences of these
datasets on the 34 dimension-6 operator coefficients in the top-specific
flavour scenario introduced in Section 2 and defined in Tables 1 and 2. The
overlaps between the different oblongs show explicitly how a given operator
may contribute to several classes of measurements. For example, ${\cal
O}_{Ht},{\cal O}_{tW},{\cal O}_{tB}$ and ${\cal O}_{Qq}^{3,1}$ contribute to
top EW observables, i.e., single-top and $W$ helicity fraction measurements,
as well as $t\bar{t}V$ observables. In addition to these observables, ${\cal
O}_{Hu},{\cal O}_{Hd},{\cal O}_{Hq}^{(3)}$ and ${\cal O}_{Hq}^{(1)}$ also
contribute to diboson, electroweak precision and Higgs observables,
exemplifying the interplay between the various data sectors.

Our analysis is based on a newly-developed public code called Fitmaker that is
to be continually updated in the future, with the goal of continuing to
provide an adaptable, flexible and extensible framework for global SMEFT fits
over the longer term. We have implemented both a fast analytical method for
linear-order fits and a Markov Chain Monte Carlo (MCMC) procedure that is
necessary to incorporate positivity priors in operator coefficients, as appear
in specific BSM scenarios. The global analysis we present here is performed at
linear order in the SMEFT operator coefficients, and makes manifest the
interplay between Higgs and top observables, as well as the electroweak and
diboson data. We discuss two possible options for the fermion flavour
structure, one assuming a flavour-universal symmetry and the other allowing
the coefficients of operators containing third-generation fermions to vary
independently through a top-specific flavour symmetry, both of which are
explicitly broken only by the dimension-6 operators that induce shifts in
Yukawa couplings. Our framework is not limited in principle to linear order,
and we illustrate in an Appendix focusing on Higgs data how the procedure can
be used for quadratic fits as well. The same framework could also be extended
to include dimension-8 operators where relevant.

The layout of our paper is as follows. Section 2 reviews the SMEFT framework
we use, and sets out the flavour-universal scenario with SU(3)5 symmetry and
the top-specific scenario with SU(2)${}^{2}\times$SU(3)5 symmetry that we
analyze. Section 3 summarizes the dataset that we use in the global fit, which
is described in more detail in Appendix A. Section 4 describes how theory
predictions are calculated, and Section 5 sets out our fitting procedure. This
is described in more detail in Appendix B, including a nested-sampling
calculational method that we illustrate in an analysis of Higgs signal
strengths to quadratic order in the dimension-6 operator coefficients. The
results from our global linear fit are presented in Section 6, where we
display results from the electroweak, Higgs and top sectors separately and in
combination. In both the flavour-universal SU(3)5 symmetric and top-specific
SU(2)${}^{2}\times$SU(3)5 scenario, we derive constraints for all individual
operators and also constraints including all dimension-6 operators and
marginalising. Our principal results are displayed in Fig. 6, with numerical
results for the SU(2)${}^{2}\times$SU(3)3 top-specific scenario presented in
Table 6. We include dedicated discussions of sensitivities in ‘Higgs-only’
operator planes that illustrates the impact of the top data, and of the
triple-gluon operator coefficient $C_{G}$. We also analyze correlations
between the coefficients of the 34 operators in the top-specific scenario and
perform a principal component analysis, identifying the most and least
constrained combinations of SMEFT operators. Applications of our analysis to
some specific BSM scenarios are presented in Section 7, including single-field
extensions of the SM and a light-stop scenario, and a survey of fits with
contributions from any combination of two, three, four or five dimension-6
operators, none of which provide any significant evidence for physics beyond
the SM. Our conclusions are summarised in Section 8.

##  2 SMEFT Framework

We describe in this Section the dimension-6 SMEFT framework that we use for
our analysis 333For some reviews of the SMEFT, see, e.g., Refs.
Willenbrock:2014bja ; Falkowski:2015fla ; Brivio:2017vri .. After reviewing
flavour symmetry in the SM, we classify accordingly the dimension-6 operators
of the SMEFT Faroughy:2020ina  and describe two flavour symmetry assumptions
that we adopt for our global fit, namely flavour-universal and top-specific
scenarios.

###  2.1 Flavour Symmetry in the Standard Model

In the absence of the Yukawa interactions, the SM Lagrangian has a U(3)5
global symmetry Gerard:1982mm ; Chivukula:1987py  associated with independent
unitary transformations among the flavour components of its five types of
fermions: the left-handed quark doublets, right-handed charge +2/3 and -1/3
quarks, left-handed lepton doublets and right-handed charged leptons. This
symmetry can be decomposed into five SU(3) flavour rotations and five U(1)
symmetries: baryon and lepton numbers, hypercharge, a Peccei-Quinn symmetry
and independent rotations of the right-handed leptons:

$\displaystyle\mathcal{G}$ $\displaystyle=SU(3)_{q}\times SU(3)_{u}\times
SU(3)_{d}\times SU(3)_{l}\times SU(3)_{e}\times U(1)^{5}\,.$ (1)

The fermion representations transform as follows under the SU(3) rotations:

$\displaystyle q^{i}$ $\displaystyle\to U_{q}^{ij}q^{j}\,,\,u^{i}\to
U_{u}^{ij}u^{j}\,,\,d^{i}\to U_{d}^{ij}d^{j}\,,\,l^{i}\to
U_{l}^{ij}l^{j}\,,\,e^{i}\to U_{e}^{ij}e^{j},$ (2)

with U${}_{x}\in$SU(3)x. However, the SM Yukawa Lagrangian

$\displaystyle\mathcal{L}_{\text{Yuk.}}=\lambda^{ij}_{d}(\bar{q}_{i}\,H)d_{j}+\lambda^{ij}_{u}(\bar{q}_{i}\,\tilde{H})u_{j}+\lambda^{ij}_{e}(\bar{l}_{i}\,H)e_{j}+\text{h.c.}$
(3)

is manifestly not invariant under this set of transformations. The down-type
Yukawa interactions break SU(2)${}_{q}\times$SU(2)d, the up-type interactions
break SU(2)${}_{q}\times$SU(2)u and the lepton Yukawa interactions break
SU(2)${}_{l}\times$SU(2)e.

One can recover the complete flavour symmetry in the presence of the Yukawa
interactions by introducing three spurions that take the places of the Yukawa
matrices and transform as follows:

$\displaystyle Y_{u}\to U_{q}\,Y_{u}\,U^{\dagger}_{u}\quad,\quad Y_{d}\to
U_{q}\,Y_{d}\,U^{\dagger}_{d}\quad,\quad Y_{e}\to
U_{l}\,Y_{e}\,U^{\dagger}_{e}\,,$ (4)

such that the Yukawa Lagrangian becomes

$\displaystyle\mathcal{L}_{\text{Yuk.}}=Y^{ij}_{d}(\bar{q}_{i}\,H)d_{j}+Y^{ij}_{u}(\bar{q}_{i}\,\tilde{H})u_{j}+Y^{ij}_{e}(\bar{l}_{i}\,H)e_{j}+\text{h.c.}\,.$
(5)

The spurions take background values that can be rotated using flavour symmetry
to a basis in which the down-type quark and lepton mass terms are diagonal and
the up-type quark mass term is diagonal up to a factor of the CKM matrix, $V$:

$\displaystyle\langle Y_{d}\rangle^{ij}=y_{d}^{ij}\propto
m_{d}^{ij},\,\,\langle Y_{e}\rangle^{ij}=y_{e}^{ij}\propto
m_{e}^{ij},\,\,\langle
Y_{u}\rangle^{ij}=(V^{\dagger}y_{u})^{ij}\propto(V^{\dagger})^{ik}m_{u}^{kj}\,.$
(6)

After electroweak symmetry breaking (EWSB), the remaining $V$ factor can be
moved to the charged-current gauge interaction terms by an independent
rotation of the left-handed up-type quark field 444The opposite convention is
sometimes used, with a diagonal up-type mass matrix before EWSB.,
$u_{L}^{i}\to(V^{\dagger})^{ik}u_{L}^{k}$.

According to the Minimal Flavour Violation (MFV) hypothesis DAmbrosio:2002vsn
, no new sources of flavour violation exist beyond the SM Yukawa couplings. It
assumes that flavour-violating effects in, e.g., higher-dimensional operators,
appear with an associated Yukawa matrix via spurion insertions. This can be
taken as a prior for the expected sizes of the operator coefficients. Often
the operators are defined to absorb a Yukawa factor determined by the spurions
but here we shall consider the coefficients of the operators that break
SU(3)5, such as those that induce shifts in the Yukawas, as explicit breakings
of the flavour symmetry, without these spurion insertions. This simplifies the
normalisation of the operators to make the interpretation of their
coefficients clear and unambiguous.

###  2.2 SMEFT operators

$X^{3}$ |  $H^{6}$ and $H^{4}D^{2}$ | $\psi^{2}H^{3}$
---|---|---
$\mathcal{O}_{\scriptscriptstyle G}$ | $f^{ABC}G_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}$ | $\mathcal{O}_{\scriptscriptstyle H}$ | $(H^{\dagger}H)^{3}$ | $\mathcal{O}_{\scriptscriptstyle eH}$ | $(H^{\dagger}H)(\bar{l}_{p}e_{r}H)$
$\mathcal{O}_{\scriptscriptstyle\widetilde{G}}$ | $f^{ABC}\widetilde{G}_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}$ | $\mathcal{O}_{\scriptscriptstyle H\Box}$ | $(H^{\dagger}H)\raisebox{-1.42262pt}{$\Box$}(H^{\dagger}H)$ | $\mathcal{O}_{\scriptscriptstyle uH}$ | $(H^{\dagger}H)(\bar{q}_{p}u_{r}\widetilde{H})$
$\mathcal{O}_{\scriptscriptstyle W}$ | $\varepsilon^{IJK}W_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}$ | $\mathcal{O}_{\scriptscriptstyle HD}$ | $\left(H^{\dagger}D^{\mu}H\right)^{\star}\left(H^{\dagger}D_{\mu}H\right)$ | $\mathcal{O}_{\scriptscriptstyle dH}$ | $(H^{\dagger}H)(\bar{q}_{p}d_{r}H)$
$\mathcal{O}_{\scriptscriptstyle\widetilde{W}}$ | $\varepsilon^{IJK}\widetilde{W}_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}$ |  |  |  |
$X^{2}H^{2}$ | $\psi^{2}XH$ | $\psi^{2}H^{2}D$
$\mathcal{O}_{\scriptscriptstyle HG}$ | $H^{\dagger}H\,G^{A}_{\mu\nu}G^{A\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle eW}$ | $(\bar{l}_{p}\sigma^{\mu\nu}e_{r})\tau^{I}HW_{\mu\nu}^{I}$ | $\mathcal{O}_{\scriptscriptstyle Hl}^{\scriptscriptstyle(1)}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\bar{l}_{p}\gamma^{\mu}l_{r})$
$\mathcal{O}_{\scriptscriptstyle H\widetilde{G}}$ | $H^{\dagger}H\,\widetilde{G}^{A}_{\mu\nu}G^{A\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle eB}$ | $(\bar{l}_{p}\sigma^{\mu\nu}e_{r})HB_{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle Hl}^{\scriptscriptstyle(3)}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,I}\,H}$})(\bar{l}_{p}\tau^{I}\gamma^{\mu}l_{r})$
$\mathcal{O}_{\scriptscriptstyle HW}$ | $H^{\dagger}H\,W^{I}_{\mu\nu}W^{I\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle uG}$ | $(\bar{q}_{p}\sigma^{\mu\nu}T^{A}u_{r})\widetilde{H}\,G_{\mu\nu}^{A}$ | $\mathcal{O}_{\scriptscriptstyle He}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\bar{e}_{p}\gamma^{\mu}e_{r})$
$\mathcal{O}_{\scriptscriptstyle H\widetilde{W}}$ | $H^{\dagger}H\,\widetilde{W}^{I}_{\mu\nu}W^{I\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle uW}$ | $(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tau^{I}\widetilde{H}\,W_{\mu\nu}^{I}$ | $\mathcal{O}_{\scriptscriptstyle Hq}^{\scriptscriptstyle(1)}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\bar{q}_{p}\gamma^{\mu}q_{r})$
$\mathcal{O}_{\scriptscriptstyle HB}$ | $H^{\dagger}H\,B_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle uB}$ | $(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\widetilde{H}\,B_{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle Hq}^{\scriptscriptstyle(3)}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,I}\,H}$})(\bar{q}_{p}\tau^{I}\gamma^{\mu}q_{r})$
$\mathcal{O}_{\scriptscriptstyle H\widetilde{B}}$ | $H^{\dagger}H\,\widetilde{B}_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle dG}$ | $(\bar{q}_{p}\sigma^{\mu\nu}T^{A}d_{r})H\,G_{\mu\nu}^{A}$ | $\mathcal{O}_{\scriptscriptstyle Hu}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\bar{u}_{p}\gamma^{\mu}u_{r})$
$\mathcal{O}_{\scriptscriptstyle HWB}$ | $H^{\dagger}\tau^{I}H\,W^{I}_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle dW}$ | $(\bar{q}_{p}\sigma^{\mu\nu}d_{r})\tau^{I}H\,W_{\mu\nu}^{I}$ | $\mathcal{O}_{\scriptscriptstyle Hd}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\bar{d}_{p}\gamma^{\mu}d_{r})$
$\mathcal{O}_{\scriptscriptstyle H\widetilde{W}B}$ | $H^{\dagger}\tau^{I}H\,\widetilde{W}^{I}_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle dB}$ | $(\bar{q}_{p}\sigma^{\mu\nu}d_{r})H\,B_{\mu\nu}$ | $\mathcal{O}_{\scriptscriptstyle Hud}$ | $i(\widetilde{H}^{\dagger}D_{\mu}H)(\bar{u}_{p}\gamma^{\mu}d_{r})$
$(\bar{L}L)(\bar{L}L)$ | $(\bar{R}R)(\bar{R}R)$ | $(\bar{L}L)(\bar{R}R)$
$\mathcal{O}_{\scriptscriptstyle ll}$ | $(\bar{l}_{p}\gamma_{\mu}l_{r})(\bar{l}_{s}\gamma^{\mu}l_{t})$ | $\mathcal{O}_{\scriptscriptstyle ee}$ | $(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$ | $\mathcal{O}_{\scriptscriptstyle le}$ | $(\bar{l}_{p}\gamma_{\mu}l_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$
$\mathcal{O}_{\scriptscriptstyle qq}^{\scriptscriptstyle(1)}$ | $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{q}_{s}\gamma^{\mu}q_{t})$ | $\mathcal{O}_{\scriptscriptstyle uu}$ | $(\bar{u}_{p}\gamma_{\mu}u_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$ | $\mathcal{O}_{\scriptscriptstyle lu}$ | $(\bar{l}_{p}\gamma_{\mu}l_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$
$\mathcal{O}_{\scriptscriptstyle qq}^{\scriptscriptstyle(3)}$ | $(\bar{q}_{p}\gamma_{\mu}\tau^{I}q_{r})(\bar{q}_{s}\gamma^{\mu}\tau^{I}q_{t})$ | $\mathcal{O}_{\scriptscriptstyle dd}$ | $(\bar{d}_{p}\gamma_{\mu}d_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle ld}$ | $(\bar{l}_{p}\gamma_{\mu}l_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
$\mathcal{O}_{\scriptscriptstyle lq}^{\scriptscriptstyle(1)}$ | $(\bar{l}_{p}\gamma_{\mu}l_{r})(\bar{q}_{s}\gamma^{\mu}q_{t})$ | $\mathcal{O}_{\scriptscriptstyle eu}$ | $(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$ | $\mathcal{O}_{\scriptscriptstyle qe}$ | $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{e}_{s}\gamma^{\mu}e_{t})$
$\mathcal{O}_{\scriptscriptstyle lq}^{\scriptscriptstyle(3)}$ | $(\bar{l}_{p}\gamma_{\mu}\tau^{I}l_{r})(\bar{q}_{s}\gamma^{\mu}\tau^{I}q_{t})$ | $\mathcal{O}_{\scriptscriptstyle ed}$ | $(\bar{e}_{p}\gamma_{\mu}e_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle qu}^{\scriptscriptstyle(1)}$ | $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{u}_{s}\gamma^{\mu}u_{t})$
|  | $\mathcal{O}_{\scriptscriptstyle ud}^{\scriptscriptstyle(1)}$ | $(\bar{u}_{p}\gamma_{\mu}u_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle qu}^{\scriptscriptstyle(8)}$ | $(\bar{q}_{p}\gamma_{\mu}T^{A}q_{r})(\bar{u}_{s}\gamma^{\mu}T^{A}u_{t})$
|  | $\mathcal{O}_{\scriptscriptstyle ud}^{\scriptscriptstyle(8)}$ | $(\bar{u}_{p}\gamma_{\mu}T^{A}u_{r})(\bar{d}_{s}\gamma^{\mu}T^{A}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle qd}^{\scriptscriptstyle(1)}$ | $(\bar{q}_{p}\gamma_{\mu}q_{r})(\bar{d}_{s}\gamma^{\mu}d_{t})$
|  |  |  | $\mathcal{O}_{\scriptscriptstyle qd}^{\scriptscriptstyle(8)}$ | $(\bar{q}_{p}\gamma_{\mu}T^{A}q_{r})(\bar{d}_{s}\gamma^{\mu}T^{A}d_{t})$
$(\bar{L}R)(\bar{R}L)$ and $(\bar{L}R)(\bar{L}R)$ |  $B$-violating
$\mathcal{O}_{\scriptscriptstyle ledq}$ | $(\bar{l}_{p}^{j}e_{r})(\bar{d}_{s}q_{t}^{j})$ | $\mathcal{O}_{\scriptscriptstyle duq}$ | $\varepsilon^{\alpha\beta\gamma}\varepsilon_{jk}\left[(d^{\alpha}_{p})^{T}Cu^{\beta}_{r}\right]\left[(q^{\gamma j}_{s})^{T}Cl^{k}_{t}\right]$
$\mathcal{O}_{\scriptscriptstyle quqd}^{\scriptscriptstyle(1)}$ | $(\bar{q}_{p}^{j}u_{r})\varepsilon_{jk}(\bar{q}_{s}^{k}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle qqu}$ | $\varepsilon^{\alpha\beta\gamma}\varepsilon_{jk}\left[(q^{\alpha j}_{p})^{T}Cq^{\beta k}_{r}\right]\left[(u^{\gamma}_{s})^{T}Ce_{t}\right]$
$\mathcal{O}_{\scriptscriptstyle quqd}^{\scriptscriptstyle(8)}$ | $(\bar{q}_{p}^{j}T^{A}u_{r})\varepsilon_{jk}(\bar{q}_{s}^{k}T^{A}d_{t})$ | $\mathcal{O}_{\scriptscriptstyle qqq}$ |  $\varepsilon^{\alpha\beta\gamma}\varepsilon_{jn}\varepsilon_{km}\left[(q^{\alpha j}_{p})^{T}Cq^{\beta k}_{r}\right]\left[(q^{\gamma m}_{s})^{T}Cl^{n}_{t}\right]$
$\mathcal{O}_{\scriptscriptstyle lequ}^{\scriptscriptstyle(1)}$ | $(\bar{l}_{p}^{j}e_{r})\varepsilon_{jk}(\bar{q}_{s}^{k}u_{t})$ | $\mathcal{O}_{\scriptscriptstyle duu}$ |  $\varepsilon^{\alpha\beta\gamma}\left[(d^{\alpha}_{p})^{T}Cu^{\beta}_{r}\right]\left[(u^{\gamma}_{s})^{T}Ce_{t}\right]$
$\mathcal{O}_{\scriptscriptstyle lequ}^{\scriptscriptstyle(3)}$ | $(\bar{l}_{p}^{j}\sigma_{\mu\nu}e_{r})\varepsilon_{jk}(\bar{q}_{s}^{k}\sigma^{\mu\nu}u_{t})$ |  |
Table 1: Dimension-6 operators in the Warsaw basis, adapted from Ref.
Grzadkowski:2010es . The grey cells indicate operators that break flavour
SU(3)5 explicitly.

The SM Lagrangian is only the leading approximation, consisting of operators
up to dimension 4, to an EFT that also includes higher-dimensional operators.
The lepton number-violating Weinberg operator Weinberg:1979sa  is the unique
one at dimension 5. At dimension 6, we may write the SMEFT Lagrangian as

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\sum_{i=1}^{2499}\frac{C_{i}}{\Lambda^{2}}\mathcal{O}_{i}\,,$
(7)

where $\Lambda$ is a dimensionful scale and $C_{i}$ are the dimensionless
Wilson coefficients. The dimension-6 operators $\mathcal{O}_{i}$ in the Warsaw
basis are given in Table 1 (adapted from Ref. Grzadkowski:2010es ). Those with
only bosonic fields are invariant under flavour symmetry, whereas those
containing fermions can be classified in terms of their transformation
properties under SU(3)5. Any off-diagonal flavour entry violates SU(3)5
explicitly, as do terms with both a left- and a right-handed field, even in
their flavour-diagonal entries. For terms involving two fermions, these
include the Yukawa ($\psi^{2}H^{3}$) and dipole ($\psi^{2}XH$) operators as
well as the right-handed charged-current operator
($\mathcal{O}_{\scriptscriptstyle Hud}$). The four-fermion sector includes the
$(\bar{L}R)(\bar{R}L)$, $(\bar{L}R)(\bar{L}R)$ and $B$-violating classes of
operators. All other operators containing fermionic currents are
SU(3)5-invariant in the flavour-universal case, i.e., when the diagonal
entries have a common Wilson coefficient and off-diagonal entries vanish.

####  2.2.1 Flavour-universal scenario

Assuming SU(3)5 symmetry reduces the Warsaw basis to the flavour-universal
scenario in which only the operators in cells not shaded grey in Table 1 are
allowed, with common flavour-diagonal Wilson coefficients and no off-diagonal
entries. Neglecting CP-violating interactions, one is left with 31 degrees of
freedom, of which 16 are relevant for a leading-order fit to electroweak
precision, diboson and Higgs data. To this we also add 4 operators that
explicitly break the flavour symmetry and affect Higgs physics through a shift
of the tau, muon, $b$-quark and top-quark Yukawa couplings. The 20 operators
in our “flavour-universal” scenario are then  EWPO:
$\displaystyle\quad\mathcal{O}_{HWB}\,,\,\mathcal{O}_{HD}\,,\,\mathcal{O}_{ll}\,,\,\mathcal{O}_{Hl}^{(3)}\,,\,\mathcal{O}_{Hl}^{(1)}\,,\,\mathcal{O}_{He}\,,\,\mathcal{O}_{Hq}^{(3)}\,,\,\mathcal{O}_{Hq}^{(1)}\,,\,\mathcal{O}_{Hd}\,,\,\mathcal{O}_{Hu}\,,\,$
Bosonic:
$\displaystyle\quad\mathcal{O}_{H\Box}\,,\,\mathcal{O}_{HG}\,,\,\mathcal{O}_{HW}\,,\,\mathcal{O}_{HB}\,,\,\mathcal{O}_{W}\,,\,\mathcal{O}_{G}\,,\,$
Yukawa: $\displaystyle\quad\mathcal{O}_{\tau H}\,,\,\mathcal{O}_{\mu
H}\,,\,\mathcal{O}_{bH}\,,\,\mathcal{O}_{tH}\,.$ (8) We have categorised these
operators roughly, into sets that are mostly constrained by electroweak
precision observables (EWPO), those that can only be constrained at tree-level
by Higgs and diboson measurements (bosonic), and operators that induce shifts
in the Yukawa couplings (Yukawa).

####  2.2.2 Top-specific flavour scenario

The minimal flavour scenario that singles out top-quark couplings relaxes the
SU(3)5 symmetry as follows AguilarSaavedra:2018nen :

$\displaystyle SU(3)^{5}$ $\displaystyle\to SU(2)^{2}\times SU(3)^{3}$
$\displaystyle=SU(2)_{q}\times SU(2)_{u}\times SU(3)_{d}\times SU(3)_{l}\times
SU(3)_{e}\,.$

This allows chirality-flipping interaction terms involving the third-
generation quark doublet and right-handed up-type fields, notably the top-
quark Yukawa interaction. The following three additional dimension-6 operators
in the SMEFT are now allowed: 555The analogous operator
$[\mathcal{O}_{\scriptscriptstyle uH}]^{\scriptscriptstyle
33}=\mathcal{O}_{\scriptscriptstyle tH}$ is already included in the set of
Yukawa operators listed in Eq. (8).

$[\mathcal{O}_{\scriptscriptstyle uG}]^{\scriptscriptstyle
33}\;=\;\mathcal{O}_{\scriptscriptstyle
tG}\,,\quad[\mathcal{O}_{\scriptscriptstyle uB}]^{\scriptscriptstyle
33}\;=\;\mathcal{O}_{\scriptscriptstyle
tB}\,,\quad[\mathcal{O}_{\scriptscriptstyle uW}]^{\scriptscriptstyle
33}\;=\;\mathcal{O}_{\scriptscriptstyle tW}\,.$ (9)

The flavour-universality conditions on operators in the $\psi^{2}H^{2}D$,
$\bar{L}L\bar{L}L$, $\bar{R}R\bar{R}R$ and $\bar{L}L\bar{R}R$ classes that
contain $q$ or $u$ are also relaxed. Schematically,

$\displaystyle
C_{\text{univ.}}\sum_{i=1,2,3}\mathcal{K}^{\mu}\bar{f}_{i}\gamma_{\mu}\,f_{i}\to\begin{cases}&C_{3}\mathcal{K}^{\mu}\bar{f}_{3}\gamma_{\mu}\,f_{3}\,,\\\
&C_{\text{univ.}}\sum_{j=1,2{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(,3)}}}}\mathcal{K}^{\mu}\bar{f}_{j}\gamma_{\mu}\,f_{j}\,,\end{cases}$
(10)

where $\mathcal{K}^{\mu}$ is a combination of other fields. Here a choice must
be made whether to split the degrees of freedom into a fully-universal
operator that preserves the full flavour symmetry or an operator that respects
only the reduced symmetry SU(2)${}^{2}\times$SU(3)3, in addition to the new
third-generation operator, as highlighted in red in (10). The two are related
by a basis rotation. We adopt the second option, since it better separates the
degrees of freedom that affect only top measurements from the rest. The
$\psi^{2}H^{2}D$ class grows to

$\displaystyle\begin{split}[\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(1)}]&\to[\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(1)}]^{j,j}\text{ and }[\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(1)}]^{3,3}=\\{\mathcal{O}_{\scriptscriptstyle
Hq_{i}}^{\scriptscriptstyle(1)},\mathcal{O}_{\scriptscriptstyle
HQ}^{\scriptscriptstyle(1)}\\}\,,\\\ [\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(3)}]&\to[\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(3)}]^{j,j}\text{ and }[\mathcal{O}_{\scriptscriptstyle
Hq}^{\scriptscriptstyle(3)}]^{3,3}=\\{\mathcal{O}_{\scriptscriptstyle
Hq_{i}}^{\scriptscriptstyle(3)},\mathcal{O}_{\scriptscriptstyle
HQ}^{\scriptscriptstyle(3)}\\}\,,\\\ [\mathcal{O}_{\scriptscriptstyle
Hu}]&\to[\mathcal{O}_{\scriptscriptstyle Hu}]^{j,j}\text{ and
}[\mathcal{O}_{\scriptscriptstyle
Hu}]^{3,3}=\\{\mathcal{O}_{\scriptscriptstyle
Hu_{i}},\mathcal{O}_{\scriptscriptstyle Ht\phantom{i}}\\}\,,\end{split}$ (11)

where $Q$ denotes here the third-generation quark doublet. Four-fermion
operators are split generically into ‘four-light’, ‘two-heavy-two-light’ and
‘four-heavy’ flavour components, where ‘light’ and ‘heavy’ denote the first
two and the third generations, respectively. The four-light degrees of freedom
are the same as the four-fermion operators of the SU(3)5 scenario, except that
they are flavour-universal over three generations for $d$, $l$ and $e$, and
only the first two generations for $q$ and $u$. The classification of the
additional four-fermion operators under this generalisation is slightly more
involved, due to the permutation symmetries on the flavour indices as well as
the equivalence of certain degrees of freedom via Fierz identities. This is
discussed in Ref. AguilarSaavedra:2018nen , where a ‘dim6top’ basis is chosen
for the operators involving top fields with LHC top physics observables in
mind. The new operators are shown in Table 2 with their definitions in terms
of the Warsaw basis coefficients.

A total of 31 new CP-conserving degrees of freedom are introduced by this
relaxation of the flavour symmetry. However, our analysis is only sensitive to
a subset of these, for two main reasons. First, our chosen dataset does not
constrain a number of the operators allowed by this flavour assumption. These
include all flavour-universal four-light fermion operators, which are
constrained by electron-positron collider data and numerous low-energy
scattering and decay experiments (see Ref. Falkowski:2017pss  for a recent
compilation of constraints), as well as high-energy Drell-Yan and dijet
observables at hadron colliders Domenech:2012ai ; Farina:2016rws ;
Dawson:2018dxp ; Alte:2018xgc ; Fuentes-Martin:2020lea ; Ricci:2020xre .
Furthermore, it is also effectively blind to the two-heavy quark two-lepton
class of operators listed in Table 2. Although many of them mediate the same
final states as those selected by, _e.g._ , $t\bar{t}V$ measurements, they do
so in the absence of a resonant intermediate $W$ or $Z$ boson decaying into
the lepton pair. So far, searches have implemented selections to enhance this
resonant contribution, and are therefore not sensitive to the non-resonant
phase space populated by the operators in question. The second important
feature of our analysis is that it is restricted to the linear,
$\mathcal{O}(1/\Lambda^{2})$, level in the EFT expansion. This restricts the
sensitivity to the set of operators that interfere appreciably with the
dominant SM amplitudes for the processes of interest. This is not the case for
the six neutral-current mediating, two-heavy two-light operators in the upper
left section of Table 2. These operators mediate $q\bar{q}\to t\bar{t}$
production in the colour-singlet channel, which does not interfere at LO with
the strongly-dominant SM QCD contribution. In contrast, the corresponding
charged-current operator affects single-top quark production that, being an EW
process in the SM, does have such an interference term. Finally, we also omit
the four-heavy operators that would mainly be constrained by
$t\bar{t}b\bar{b}$ measurements and four-top production searches. These data
have been shown to be largely sensitive, at present, to the quadratic EFT
contributions Degrande:2010kt ; DHondt:2018cww ; Hartland:2019bjb ;
Banelli:2020iau , and our analysis would not yield meaningful bounds in these
directions. We therefore only include in our analysis 8 two-heavy two-light
quark degrees of freedom: the colour-singlet, charged-current operator and
seven neutral, colour-octet operators.

To summarise, the 34 operators relevant for our leading-order, linear fit in
the top-specific flavour scenario are the 20 listed in Eqs. (8) plus the 14
discussed above (three in (9), three more in (11) and eight two-light two-
heavy quark operators):  EWPO:
$\displaystyle\quad\mathcal{O}_{HWB}\,,\,\mathcal{O}_{HD}\,,\,\mathcal{O}_{ll}\,,\,\mathcal{O}_{Hl}^{(3)}\,,\,\mathcal{O}_{Hl}^{(1)}\,,\,\mathcal{O}_{He}\,,\,\mathcal{O}_{Hq}^{(3)}\,,\,\mathcal{O}_{Hq}^{(1)}\,,\,\mathcal{O}_{Hd}\,,\,\mathcal{O}_{Hu}\,,\,$
Bosonic:
$\displaystyle\quad\mathcal{O}_{H\Box}\,,\,\mathcal{O}_{HG}\,,\,\mathcal{O}_{HW}\,,\,\mathcal{O}_{HB}\,,\,\mathcal{O}_{W}\,,\,\mathcal{O}_{G}\,,\,$
Yukawa: $\displaystyle\quad\mathcal{O}_{\tau H}\,,\,\mathcal{O}_{\mu
H}\,,\,\mathcal{O}_{bH}\,,\,\mathcal{O}_{tH}\,,$ Top 2F:
$\displaystyle\quad\mathcal{O}_{HQ}^{(3)}\,,\,\mathcal{O}_{HQ}^{(1)}\,,\,\mathcal{O}_{Ht}\,,\,\mathcal{O}_{tG}\,,\,\mathcal{O}_{tW}\,,\,\mathcal{O}_{tB}\,,$
Top 4F:
$\displaystyle\quad\mathcal{O}_{Qq}^{3,1}\,,\,\mathcal{O}_{Qq}^{3,8}\,,\,\mathcal{O}_{Qq}^{1,8}\,,\,\mathcal{O}_{Qu}^{8}\,,\,\mathcal{O}_{Qd}^{8}\,,\,\mathcal{O}_{tQ}^{8}\,,\,\mathcal{O}_{tu}^{8}\,,\,\mathcal{O}_{td}^{8}\,.$
(12) These are grouped into top operators involving two (top 2F) and four (top
4F) heavy fermions, respectively.

$\mathcal{O}_{i}$ |  $C_{i}$ Definition |  $\mathcal{O}_{i}$ |  $C_{i}$ Definition
---|---|---|---
_4 quark (2 heavy 2 light)_
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}\left([C_{qq}^{(1)}]^{\scriptscriptstyle ii33}+\frac{1}{6}[C_{qq}^{(1)}]^{\scriptscriptstyle i33i}+\frac{1}{2}[C_{qq}^{(3)}]^{\scriptscriptstyle i33i}\right)$ |  $\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 1,8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}\left([C_{qq}^{(1)}]^{\scriptscriptstyle i33i}+3[C_{qq}^{(3)}]^{\scriptscriptstyle i33i}\right)$
$\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}\left([C_{qq}^{(3)}]^{\scriptscriptstyle ii33}+\frac{1}{6}[C_{qq}^{(1)}]^{\scriptscriptstyle i33i}-\frac{1}{6}[C_{qq}^{(3)}]^{\scriptscriptstyle i33i}\right)$ |  $\mathcal{O}_{\scriptscriptstyle Qq}^{\scriptscriptstyle 3,8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}\left([C_{qq}^{(1)}]^{\scriptscriptstyle i33i}-[C_{qq}^{(3)}]^{\scriptscriptstyle i33i}\right)$
$\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}\left([C_{uu}]^{\scriptscriptstyle ii33}+\frac{1}{3}[C_{uu}]^{\scriptscriptstyle i33i}\right)$ |  $\mathcal{O}_{\scriptscriptstyle tu}^{\scriptscriptstyle 8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}2[C_{uu}]^{\scriptscriptstyle i33i}$
$\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2(,3)}[C_{ud}^{(1)}]^{\scriptscriptstyle 33ii}$ |  $\mathcal{O}_{\scriptscriptstyle td}^{\scriptscriptstyle 8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2(,3)}[C_{ud}^{(8)}]^{\scriptscriptstyle 33ii}$
$\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}[C_{qu}^{(1)}]^{\scriptscriptstyle ii33}$ |  $\mathcal{O}_{\scriptscriptstyle tq}^{\scriptscriptstyle 8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}[C_{qu}^{(8)}]^{\scriptscriptstyle ii33}$
$\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}[C_{qu}^{(1)}]^{\scriptscriptstyle 33ii}$ |  $\mathcal{O}_{\scriptscriptstyle Qu}^{\scriptscriptstyle 8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2}[C_{qu}^{(8)}]^{\scriptscriptstyle 33ii}$
$\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 1}$ |  $\sum\limits_{\scriptscriptstyle i=1,2(,3)}[C_{qd}^{(1)}]^{\scriptscriptstyle 33ii}$ |  $\mathcal{O}_{\scriptscriptstyle Qd}^{\scriptscriptstyle 8}$ |  $\sum\limits_{\scriptscriptstyle i=1,2(,3)}[C_{qd}^{(8)}]^{\scriptscriptstyle 33ii}$
_4 quark (4 heavy)_
$\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 1}$ |  $2[C_{qq}^{(1)}]^{\scriptscriptstyle 3333}-\frac{2}{3}[C_{qq}^{(3)}]^{\scriptscriptstyle 3333}$ |  $\mathcal{O}_{\scriptscriptstyle QQ}^{\scriptscriptstyle 8}$ |  $8[C_{qq}^{(3)}]^{\scriptscriptstyle 3333}$
$\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 1}$ |  $[C_{qu}^{(1)}]^{\scriptscriptstyle 3333}$ |  $\mathcal{O}_{\scriptscriptstyle Qt}^{\scriptscriptstyle 8}$ |  $[C_{qu}^{(8)}]^{\scriptscriptstyle 3333}$
$\mathcal{O}_{\scriptscriptstyle tt}$ |  $[C_{uu}^{(1)}]^{\scriptscriptstyle 3333}$ |  |
_2 heavy 2 lepton_
$\mathcal{O}_{\scriptscriptstyle Ql}^{\scriptscriptstyle-(1)}$ |  $\sum\limits_{\scriptscriptstyle i=1,2,3}[C_{lq}^{1}]^{\scriptscriptstyle ii33}-[C_{lq}^{3}]^{\scriptscriptstyle ii33}$ |  $\mathcal{O}_{\scriptscriptstyle tl}^{\scriptscriptstyle(1)}$ |  $\sum\limits_{\scriptscriptstyle i=1,2,3}[C_{lu}]^{\scriptscriptstyle ii33}$
$\mathcal{O}_{\scriptscriptstyle Ql}^{\scriptscriptstyle 3(1)}$ |  $\sum\limits_{\scriptscriptstyle i=1,2,3}[C_{lq}^{3}]^{\scriptscriptstyle ii33}$ |  $\mathcal{O}_{\scriptscriptstyle te}^{\scriptscriptstyle(1)}$ |  $\sum\limits_{\scriptscriptstyle i=1,2,3}[C_{eu}]^{\scriptscriptstyle ii33}$
$\mathcal{O}_{\scriptscriptstyle Qe}^{\scriptscriptstyle(1)}$ |  $\sum\limits_{\scriptscriptstyle i=1,2,3}[C_{eQ}]^{\scriptscriptstyle ii33}$ |  |
Table 2:  Four-fermion operators containing at least one third-generation
bilinear in the ‘dim6top’ basis AguilarSaavedra:2018nen  assuming an
SU(2)${}^{2}\times$SU(3)3 flavour symmetry. The relations of the corresponding
Wilson coefficients with those of the Warsaw basis are also shown. The shaded
entries indicate operators that are not included in our analysis because
significant constraints cannot be obtained from the chosen dataset at leading
order and linear level in the EFT expansion, as discussed in the text.

##  3 Dataset description

In this Section we describe the data used in our global fit. We summarise here
the main categories of data and refer the reader to Appendix A for a complete
list of the observables that have been implemented in Fitmaker, together with
their source references.

The most precise electroweak measurements, other than the $W$ mass, remain
those from LEP and the SLC 666We note that global SMEFT fits would benefit
greatly from a future $Z$-pole run Ellis:2015sca ; deBlas:2019wgy ;
Abada:2019zxq .. The Higgs boson discovery at the LHC enabled the possibility
of a closed global SMEFT fit to a complete set of dimension-6 operators for
the first time. Higgs physics has since progressed rapidly to include more
channels and sub-categories beyond signal strengths. In particular, the STXS
categorisations of the various Higgs production sub-channels provide further
sensitivity to different directions in the fit, as illustrated, for example,
in Fig. 2 below for the case of gluon fusion and described further in the next
section.

The higher energies at the LHC also allow certain measurements of diboson and
dilepton final states to become competitive with LEP Zhang:2016zsp ;
Grojean:2018dqj ; Farina:2016rws ; Dawson:2018dxp ; Alte:2018xgc ; Fuentes-
Martin:2020lea ; Ricci:2020xre , enable complementary probes of higher-
dimensional operators Henning:2018kys ; Falkowski:2020znk , and, moreover,
give access to top physics with higher statistics than ever before, including
the previously unreachable $t\bar{t}W/Z/H$ and other, rare production
processes such as four-top production Degrande:2010kt ; DHondt:2018cww ;
Hartland:2019bjb ; Banelli:2020iau . More operators, under less restrictive
flavour assumptions, can then be included in a global SMEFT fit. This is
particularly motivated since the top quark is often expected to be more
sensitive to BSM physics.

The following is a summary of the different categories of observables that we
consider—see Appendix A for more details and references. We build our selected
dataset by combining statistically independent measurements, including
correlation information by means of published covariance/correlation matrices,
when available 777See Ref. Bissmann:2019qcd  for a study of the impact of
correlations in global fits.. In general, for LHC data, this amounts to a
single ATLAS and CMS measurement in a particular final state for each LHC run.
When multiple measurements, e.g., differential distributions, are reported, a
single one is chosen based on maximising the sensitivity of our fit.

  * •

The set of electroweak precision observables (EWPOs) include the pseudo-
observables measured on the $Z$ resonance by LEP and SLD, together with the
$W$ boson mass measurements by CDF and D0 at the Tevatron and ATLAS at the
LHC:

$\displaystyle\text{EWPO:
}\quad\\{\Gamma_{Z},\sigma^{0}_{\text{had.}},R_{l}^{0},A_{FB}^{l},A_{l},R_{b}^{0},R_{c}^{0},A_{FB}^{b},A_{FB}^{c},A_{b},A_{c},M_{W}\\}\,.$
(13)

We include a total of 14 electroweak measurements.

  * •

For diboson measurements, we focus on the $W^{+}W^{-}$ measurements of total
cross-sections at different energies and angular distributions at LEP, and the
differential distribution for the fiducial cross-section binned in leading
lepton $p_{T}$ by ATLAS at the LHC. We also incorporate the differential
distribution in $\Delta\varphi_{jj}$ for the $Zjj$ measurement given by ATLAS,
that we include in the diboson category due to the related physics to which it
is sensitive. We include a total of 100 diboson measurements.

  * •

The Higgs dataset at the LHC include the combination of Higgs signal strengths
by ATLAS and CMS for Run 1, and for Run 2 both signal strengths and STXS
measurements are used. ATLAS in particular provide the combined stage $1.0$
STXS for $4l,\gamma\gamma,WW^{*},\tau^{+}\tau^{-}$ and $b\bar{b}$, while for
CMS we use the signal strengths of $4l$, $\gamma\gamma$, $WW^{*}$,
$\tau^{+}\tau^{-}$, $b\bar{b}$ and $\mu^{+}\mu^{-}$. We also include the
$Z\gamma$ signal strength from ATLAS and a differential $WW^{*}$ cross-section
measurement from CMS. We include a total of 72 Higgs measurements.

  * •

The top data consists of differential distributions in various $t\bar{t}$
channels and cross-section measurements of top pair production in association
with a $W/Z$ gauge boson (the $t\bar{t}V$ dataset), as well as various single
top differential and cross-section measurements, for both Runs 1 and 2. We
include a total of 124 top measurements.

Overall, we include a total of 310 measurements in our analysis.

If not already given in such a form, each measurement is converted into a
corresponding ‘signal strength’, $\mu$, defined as the ratio of the observed
value to the best available theory prediction, usually quoted in the
experimental publication. Differential data is taken from the publication and
its associated entry in HEPdata, where available, using absolute differential
cross section measurements, $\vec{\sigma}_{\text{abs}}$, and their associated
covariance matrices, $\mathbf{\Sigma}_{\text{abs}}$. If only normalised
differential cross sections ($\vec{\sigma}_{\text{norm}}$) are published, they
are converted to absolute ones using the best available measurement of the
inclusive cross section for that process in the same channel. Covariance
matrices are then updated to reflect the correlations between the bins induced
by the common rescaling of the total cross section,
$\sigma_{\text{tot}}\pm\delta\sigma_{\text{tot}}$. The absolute differential
measurement and its covariance matrix are then

$\displaystyle\vec{\sigma}_{\text{abs}}$
$\displaystyle=\vec{\sigma}_{\text{norm}}\sigma_{\text{tot}}\,,$
$\displaystyle\mathbf{\Sigma}_{\text{abs}}$
$\displaystyle=\mathbf{\Sigma}_{\text{norm}}\sigma_{\text{tot}}^{2}+\delta\sigma_{\text{tot}}^{2}\vec{\sigma}_{\text{norm}}\otimes\vec{\sigma}_{\text{norm}}\,.$

Where available, `fastnlo` tables Czakon:2017dip ; Czakon:2019bcq ;
Czakon:2019yrx  were used to obtain NNLO QCD predictions for the differential
$t\bar{t}$ data. The SM theoretical errors are taken to be uncorrelated and
the relative signal strength covariance matrix is obtained by adding the
relative experimental and theoretical covariances as follows:

$\mathbf{\Sigma}_{\mu}=\left(\frac{\mathbf{\Sigma}_{\text{exp}}}{\vec{\sigma}_{\text{exp}}\otimes\vec{\sigma}_{\text{exp}}}+\text{diag}(\vec{\delta}_{\text{th}}/\vec{\sigma}_{\text{th}})^{2}\right)(\vec{\mu}\otimes\vec{\mu}),\;\;{\rm
where}\;\;\;\vec{\mu}\equiv\frac{\vec{\sigma}_{\text{exp}}}{\vec{\sigma}_{\text{th}}}\,.$
(14)

This corresponds to adding the relative experimental and theory uncertainties
in quadrature. The observables are stored in the `Fitmaker` database in `json`
format, together with metadata and information about how each signal strength
was obtained.

##  4 SMEFT Predictions

###  4.1 General strategy

The SMEFT predictions for all of the included observables were computed using
the code MadGraph5_aMC@NLO together with the SMEFTsim Brivio:2017btx  and/or
the SMEFT@NLO Degrande:2020evl  UFO models. These are used to extract the
linear contribution, $a_{i}^{\scriptscriptstyle X}$, of a given Wilson
coefficient, $C_{i}$, to a physical quantity, $X$, such as a production cross-
section, partial or total decay width, or asymmetry:

$\displaystyle\mu_{\scriptscriptstyle
X}\equiv\frac{X}{X_{SM}}=1+\sum_{i}a_{i}^{\scriptscriptstyle
X}\frac{C_{i}}{\Lambda^{2}}+\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right).$
(15)

The $a^{X}_{i}$ can usually be obtained with a single, high statistics Monte-
Carlo (MC) run for each coefficient. In some cases, non-linear contributions
from Wilson coefficients can arise in MC predictions due to the $W$-mass shift
and modifications of total widths of intermediate particles. The former is a
consequence of using the EW $\\{\alpha_{\scriptscriptstyle
EW},G_{\scriptscriptstyle F},m_{\scriptscriptstyle Z}\\}$ input scheme, while
the latter is a feature of the physical prediction for the observable, and can
often be factorised into production and decay via the narrow-width
approximation (NWA), when a narrow resonance such as the Higgs, $W$, $Z$ or
top is produced on-shell. This factorisation allows the decay to be computed
separately then added to the prediction. Failing that, the $a^{X}_{i}$ are
obtained by fitting polynomial functions to the observables’ dependences on a
given coefficient. This is done, in particular, for Higgs production in
association with a $W$-boson (WH), vector boson fusion (VBF), and $pp\to
WW\to\ell\nu\ell\nu$.

Predictions are obtained at Leading Order (LO) in perturbation theory
throughout, which, in almost all cases, corresponds to tree-level
computations. Unless stated otherwise, we generate and analyse our events at
the parton level and apply analysis-specific selection criteria to obtain the
relevant fiducial regions of phase space. We use the following values in the
aforementioned EW input scheme:

$\begin{gathered}\alpha_{\scriptscriptstyle EW}^{-1}=127.95,\quad
G_{\scriptscriptstyle F}=1.16638\times 10^{-5}\,\text{GeV}^{-2}\,,\\\
m_{\scriptscriptstyle Z}=91.1876\,\text{GeV},\quad m_{\scriptscriptstyle
H}=125.09\,\text{GeV},\quad m_{t}=173.2\,\text{GeV}\,.\end{gathered}$ (16)

All other fermions are taken to be massless, which implies the use of five-
flavour scheme PDFs, for which we use the default NNPDF23_nlo_as_0119 sets
Ball:2012cx  provided by MadGraph5_aMC@NLO. The one exception is when the
lighter fermions appear as Higgs boson decay products, when we assume that
they interact with the Higgs via their Yukawa couplings, $y_{f}\equiv
2^{\frac{3}{2}}m_{f}\sqrt{G_{\scriptscriptstyle F}}$, with masses taken to be

$\displaystyle m_{\mu}=0.106\,\text{GeV}\,,\quad
m_{\tau}=1.77\,\text{GeV},\quad m_{c}=0.907\,\text{GeV},\quad\text{and}\quad
m_{b}=3.237\,\text{GeV}\,.$ (17)

The latter two have been run up to the Higgs mass scale. In some cases, the
3rd generation-specific operators independently modify $b$-quark initiated
contributions to EW processes such as VBF and diboson. We do not take these
contributions into account as they are highly suppresed by the $b$ PDFs. We do
not assign a theory uncertainty to our predictions of the SMEFT contributions,
assuming that they will be subdominant with respect to other uncertainties
such as the baseline SM theory predictions. We also neglect other theoretical
uncertainties inherent to the SMEFT framework itself, such as omitting
quadratic dimension-6 or linear dimension-8 contributions and other higher-
order effects (see, e.g., Refs. Englert:2014cva ; Hays:2020scx ; Horne:2020pot
; Keilmann:2019cbp ; Hays:2018zze ; Baglio:2020oqu  for discussions of these
and related uncertainties). We note that operator mixing from RGE running and
loops can also induce extra constraints Cirigliano:2016nyn ; Aoude:2020dwv ;
Bissmann:2019gfc  and the effects of including SMEFT operators in parton
distribution functions are also starting to be investigated Carrazza:2019sec .

###  4.2 Higgs production

We computed LO predictions in the full parameter space of our basis for the
five main Higgs production modes: gluon-fusion (ggF), VBF, associated
production with a $W$ or $Z$ ($WH,ZH$), and associated production with a top
quark pair, $t\bar{t}H$. Some results are taken from Ref. Ellis:2018gqa ,
after being cross checked by independent computations. The only one-loop
calculations that we employ in our analysis involve the Higgs coupling to
gluons, for which the LO contribution in the SM arises at one-loop level.
Despite being a loop-induced coupling it mediates the $gg\to H$ Higgs boson
production mode, which is dominant at hadron colliders. Since many Higgs
measurements are very sensitive to this production mechanism and the
associated, $gg\to H(+\text{jets})$ processes, we include as leading effects
in the SMEFT both the tree-level contribution from $C_{HG}$ and the leading
effects from the operators that modify the top-loop contribution to the SM
coupling: $C_{tH}$, $C_{tG}$, and $C_{H\Box}$. One final operator, $C_{G}$,
modifies the gluon self-interaction allowing for a contribution to gluon
fusion Higgs production in association with one or more jets, which we also
include at one-loop order in this channel for the first time. The computations
of linear contributions involve extracting the interference between loop
diagrams of the SM with tree-level diagrams from $C_{HG}$ as well as loop
diagrams with a single operator insertion, and are made possible by SMEFT@NLO.
We use a fixed renormalisation and factorisation scale of
$m_{\scriptscriptstyle H}$ in all such computations.

Figure 2 illustrates the predictions we obtain for a selection of stage 1.1
STXS gluon fusion bins Berger:2019wnu , highlighting the additional
discriminating power offered by the inclusion of $H$+jet(s). These predictions
were obtained by parton-level generation of Higgs production in association
with one or two additional jets. Specific predictions for, e.g., the 0-jet
gluon-fusion bin with Higgs $p_{T}$ > 10 GeV or the ‘3-jet like’ $\geq 2$-jet
bins with $p_{T}(Hjj)>25$ GeV would require a matching/merging procedure
interfaced with parton showering that goes beyond the level of sophistication
of our analysis. Instead, we assume the same dependence on the coefficients as
we find for the associated parton-level bin ($p_{T}^{H}=0$ and
$p_{T}(Hjj)=0$), assuming that the main effect on the population of the non-
zero $p_{T}$ bins will come from the parton shower, which does not depend on
the EFT coefficients.

Figure 2: Illustration of the effects of selected SMEFT operators on
representative gluon-fusion simplified template cross-sections $\sigma_{i}$
relative to the corresponding SM cross-sections $\sigma_{SM}$, for
$C_{i}/\Lambda=1$ TeV-2. The various Higgs $p_{T}$ and $m_{jj}$ bins provide
complementary sensitivities and hence discriminating power between the
operators.

If one were to use only Higgs signal strength data, they would be limited to
the sensitivity shown in the left-most ‘0-jet’ entry, and measurements of
gluon fusion would exhibit a degeneracy in the four relevant coefficients.
Instead, allowing for associated production with jets offers sensitivity to
$C_{G}$ and, most importantly, breaks the aforementioned degeneracy by
exploiting the different energy dependences of the five operators.

Another relevant loop-induced process in Higgs production is $gg\to ZH$,
which, while being formally an NNLO QCD contribution to $ZH$, accounts for a
significant portion of the inclusive cross section Brein:2003wg , and has a
harder $p_{T}$ spectrum, which could be especially relevant for EFT
interpretations Mimasu:2015nqa . Recent STXS definitions include dedicated
bins for this contribution, although we are not aware of any explicit
extraction of the cross sections. Notably, this process was shown to be
sensitive to top quark interactions with the $Z$, which are relatively poorly
measured elsewhere Bylund:2016phk ; Englert:2016hvy . However, we leave to
future work the inclusion of this loop-induced sensitivity to EW top
operators, also present in other processes such as $gg\to VV$ Azatov:2016xik .

###  4.3 Gauge boson and Higgs decays

Many of the measurements included in our fit involve the on-shell production
and decay of SM gauge bosons and/or the Higgs boson. As mentioned above, we
employ the NWA to factorise production and decay, such that, at the linear
level in the SMEFT, the modification to the cross section of a given process
is a combination of the $a^{X}_{i}$ for production, the partial width to the
decay channel and the total width of the parent particle. The total width
shifts of the $W$ and $Z$ bosons, have been determined as follows, where
$\Lambda$ has been fixed to 1 TeV:

$\displaystyle\begin{split}\frac{\Gamma_{Z}}{\Gamma_{Z}^{SM}}&=1-0.05\,C_{HWB}-0.041\,C_{HD}+0.082\,C_{ll}-0.13\,C_{Hl}^{(3)}-0.012\,C_{Hl}^{(1)}-0.012\,C_{He}\\\
&+0.077\,C_{Hq}^{(3)}+0.0078\,C_{Hq}^{(1)}+0.016\,C_{Hu}-0.012\,C_{Hd}+0.021\,C_{HQ}^{(3)}+0.021\,C_{HQ}^{(1)}\,,\end{split}$
(18) $\displaystyle\frac{\Gamma_{W}}{\Gamma_{W}^{SM}}$
$\displaystyle=1-0.14\,C_{HWB}-0.065\,C_{HD}+0.10\,C_{ll}-0.16\,C_{Hl}^{(3)}+0.081\,C_{Hq}^{(3)}\,.$
(19)

Higgs boson decays present a richer structure, due to the importance of four-
fermion decay modes that are mediated in the SM by the $H$ couplings to the
$W$ and $Z$. Although the Higgs mass is too small for both gauge bosons to be
on-shell, it is often assumed that at least one of the gauge bosons is,
simplifying the decay to a three-body, $H\to Vf^{\prime}\bar{f}$, final state
with the NWA applied to on-shell vector boson, $V=W,Z$. This neglects certain
interference effects between, e.g., neutral-current and charged-current
mediated four-fermion decays. The SMEFT introduces tree-level $H\gamma\gamma$,
$H\gamma Z$ and $Hgg$ interactions, all of which can contribute to four-
fermion decay modes, and degrade the accuracy of the NWA. In practice,
however, experiments often make invariant mass cuts around
$m_{\scriptscriptstyle Z}$ in, e.g., $h\to 4\ell$ analyses, that could largely
mitigate this effect.

Ref. Brivio:2019myy  performs an in-depth calculation and analysis of Higgs
decays to four fermions beyond the NWA, from which we take our predictions. We
also include the contributions to $h\to gg$ mediated by operators that modify
the top-quark loop contribution. These have the same relative impact as they
do on $gg\to H$, discussed in the previous section. The results of Ref.
Brivio:2019myy  are given for the flavour-universal $SU(3)^{5}$ scenario,
which is more restrictive than our top-specific one, singling out operators
affecting the left-handed $b$-quark couplings to the $Z$, $C_{HQ}^{(1)}$ and
$C_{HQ}^{(3)}$. In order to account for four-fermion decays involving $b$
quarks, it was necessary to adapt these results, complemented with some
predictions from SMEFT@NLO. The contributions to the total Higgs width were
found to be

$\displaystyle\begin{split}\frac{\Gamma_{H}}{\Gamma_{H}^{SM}}&=1+0.18\,C_{HWB}+0.018\,C_{HD}+0.045\,C_{ll}+0.11\,C_{H\Box}+1.7\,C_{HG}-0.075\,C_{HW}\\\
&-0.093\,C_{HB}-0.089\,C_{Hl}^{(3)}-0.000059\,C_{Hl}^{(1)}-0.000039\,C_{He}-0.000051\,C_{Hd}\\\
&+0.0037\,C_{Hq}^{(3)}-0.00025C_{Hq}^{(1)}+0.000055\,C_{Hu}-0.00014\,C_{HQ}^{(3)}+0.00038\,C_{HQ}^{(1)}\\\
&-0.73\,C_{\tau H}-0.0057\,C_{tH}-4.0\,C_{bH}-0.043\,C_{\mu
H}+0.044\,C_{tG}\,.\end{split}$ (20)

In any case, the only experimentally accessible four-fermion decay modes are
the leptonic ones, so quark current operators are only practically relevant
via their effect on the total width, which is clearly very small. For
comparison, we computed the Higgs decays using the NWA for $h\to VV^{\ast}\to
4f$ processes, noting that the two approaches can have significantly different
predictions for some operators. However, we do not find any appreciable
differences in the results of the global fit, indicating that Higgs decays are
not the primary source of constraints for the operators whose predictions are
sensitive to whether or not this approximation is used. Finally, we note that
a recent EFT interpretation of Higgs production measurements in the $h\to
4\ell$ channel Aad:2020mkp  pointed out significant acceptance differences in
the kinematical selection between the SM and the EFT due to the additional
off-shell photon contributions from $C_{HW}$, $C_{HB}$ and $C_{HWB}$. We do
not take any acceptance corrections into account in our analysis, but also do
not expect these to have a large impact on the global fit, given the fact that
our results are unaffected by whether or not we use the NWA.

###  4.4 Diboson data

We have obtained predictions for the fiducial signal strengths in bins of
$p_{T}$ for the ATLAS $WW$ analysis Aaboud:2019nkz , and for the signal
strengths in bins of $\Delta\varphi_{jj}$ for the ATLAS $Zjj$ analysis
Aad:2020sle , following the general strategy outlined above. For the $WW$
analysis, we find good agreement of the total fiducial cross-section after
implementing the analysis cuts, whereas in the $Zjj$ case we have validated
our analysis by comparison with the binned signal strengths for the operator
coefficients reported by ATLAS. In particular, as pointed out in Ref.
Aad:2020sle , we find that this channel is the most sensitive to the
interference term that is linear in the triple gauge boson operator
$\mathcal{O}_{W}$.

For LEP we used the $WW$ results of Ref. Berthier:2016tkq  that are provided
for the total and differential cross-sections at different centre of mass
energies. We note that this analysis uses a restricted set of angular
distribution bins to mitigate the effects of unknown correlations in those
bins.

###  4.5 Top data

Since we only make use of parton-level unfolded data, all top predictions are
generated with stable top quarks. The data assume SM-like decay chains for the
top quarks, and we do not take into account the small modifications to the $W$
boson branching fractions in top decays due to current operators. For top
production in association with a gauge or Higgs boson, we do take into account
the modified branching fraction of the associated boson in the measured decay
channel (usually leptonic for $W$ and $Z$). However, this choice is not
expected to have a significant impact on the results, given that top data is
mainly sensitive to top-related operators, with those that modify gauge or
Higgs boson decays being better constrained elsewhere. Much of the data we use
overlaps with the data used in the top sector fit of Ref. Hartland:2019bjb ,
in which predictions are also obtained with SMEFT@NLO, but including parton
shower effects. We make use of the linear, LO parts of these in our work 888We
thank the authors of Ref. Hartland:2019bjb  for sharing the predictions with
us. and generate new, parton-level samples for these measurements, as
including parton showering was not found to be very significant in determining
the $a^{X}_{i}$. Our work also includes some top quark asymmetry measurements,
namely those of the forward-backward and charge asymmetries from the Tevatron
and LHC, respectively. Splitting a measured cross section, $\sigma$, into two
regions, $\sigma_{F}$ and $\sigma_{B}$, the asymmetry, $A$, is defined as
$A\equiv(\sigma_{F}-\sigma_{B})/(\sigma_{F}+\sigma_{B})$. The linearised
contribution of a given operator is proportional to the difference between its
relative contributions to the cross sections in the two regions,
$a^{\scriptscriptstyle\sigma_{F}}_{i}$ and
$a^{\scriptscriptstyle\sigma_{B}}_{i}$,

$\displaystyle
a^{i}_{A}=\frac{1-A_{SM}^{2}}{2A_{SM}}(a_{i}^{\scriptscriptstyle\sigma_{F}}-a_{i}^{\scriptscriptstyle\sigma_{B}})\,,$
(21)

highlighting how asymmetry measurements may be useful in breaking parameter
space degeneracies in total cross section measurements that would, instead, be
sensitive to the sum,
$a_{i}^{\scriptscriptstyle\sigma_{F}}+a_{i}^{\scriptscriptstyle\sigma_{B}}$.
We use the latest NNLO QCD + NLO EW theory predictions for the top asymmetries
from Ref. Czakon:2017lgo . The impact of the asymmetries is quantified in
Section 6.2.

##  5 Fitting procedure

We perform a $\chi^{2}$ fit for a vector of observables, $\vec{y}$, with
covariance matrix 999In the case of a non-symmetric covariance matrix
$\bf{\tilde{V}}$, we symmetrise it by defining ${\bf
V}^{-1}=\frac{1}{2}\left({\bf\tilde{V}}^{-1}+({\bf\tilde{V}}^{-1})^{T}\right)$.
We have verified in a numerical fit to Higgs data that this does not affect
significantly the results. , ${\bf V}$, and theory predictions for those
observables, $\vec{\mu}(C_{i})$, using a $\chi^{2}$ function defined as

$\chi^{2}(C_{i})=\left(\vec{y}-\vec{\mu}(C_{i})\right)^{T}{\bf
V}^{-1}\left(\vec{y}-\vec{\mu}(C_{i})\right)\,.$ (22)

The predictions are functions of the dimension-6 operator coefficients
$C_{i}$, as defined in Eq. (7), and are truncated at the linear level so as to
include only the interference term with the SM. The quadratic dependence on
$C_{i}$ is generically of the same order as linear interference terms with
coefficients of dimension-8 operators, though exceptions exist in some
specific UV completions Liu:2016idz . The importance of their effect is
therefore a model-dependent question. We note also that sensitivity of a
linear fit to quadratic contributions is an indicator for a possible breakdown
in the regime of validity of the SMEFT, so care must be taken in the
interpretation of the fit 101010This issue is illustrated in the context of
Higgs measurements in Appendix B..

The least-squares estimators $\hat{C}_{i}$ that extremise the $\chi^{2}$
function can be obtained analytically in the case of a linear fit (see, e.g.,
Ref. Zyla:2020zbs  for a review). We may write the linear theory prediction in
terms of a matrix ${\bf H}$ that characterises the modification of the SM
predictions $\vec{\mu}^{\text{SM}}$ at linear order:

$\displaystyle\mu_{\alpha}(C_{i})=\mu^{\text{SM}}_{\alpha}+{\bf H}_{\alpha
i}C_{i}\,.$ (23)

A summation over repeated indices is implied; the index $\alpha$ ranges over
the number of observables and $i$ ranges over the number of dimension-6
coefficients. Solving $\partial\chi^{2}/\partial C_{i}=0$ gives the best fit
values as

$\hat{\vec{C}}=\left({\bf H}^{T}{\bf V}^{-1}{\bf H}\right)^{-1}{\bf H}^{T}{\bf
V}^{-1}(\vec{y}-\vec{\mu}^{\text{SM}})\equiv{\bf F}^{-1}\vec{\omega}\,.$ (24)

It is convenient to define the symmetric Hessian matrix ${\bf F}$, also known
as the Fisher information matrix, and the $\chi^{2}$ gradient vector
$\vec{\omega}$ as

${\bf F}\equiv{\bf H}^{T}{\bf V}^{-1}{\bf H}\quad,\quad\vec{\omega}\equiv{\bf
H}^{T}{\bf V}^{-1}(\vec{y}-\vec{\mu}^{\text{SM}})\,,$ (25)

in terms of which the $\chi^{2}$ function Eq. (22) can be written as

$\chi^{2}(C_{i})=\chi^{2}_{\text{SM}}-2\vec{C}^{T}\vec{\omega}+\vec{C}^{T}{\bf
F}\vec{C}\,.$ (26)

Splitting the coefficients into $\vec{C}=\\{\vec{C}_{A},\vec{C}_{B}\\}$, we
may profile over a subset of coefficients $\vec{C}_{A}$ to obtain the least-
squares estimators $\hat{\vec{C}}_{B}$ for the remaining coefficients
$\vec{C}_{B}$. For this purpose, the Fisher information matrix may be
decomposed into the sub-matrices

${\bf F}=\left(\begin{matrix}{\bf F}_{A}&{\bf F}_{AB}\\\ {\bf F}_{AB}^{T}&{\bf
F}_{B}\end{matrix}\right)\,,$ (27)

and the gradient vectors as
$\vec{\omega}=\\{\vec{\omega}_{A},\vec{\omega}_{B}\\}$. The profiled best fit
values are then given by

$\hat{\vec{C}}_{B}=\left({\bf F}_{B}-{\bf F}_{AB}^{T}{\bf F}_{A}^{-1}{\bf
F}_{AB}\right)^{-1}\left(\vec{\omega}_{B}-{\bf F}_{AB}^{T}{\bf
F}_{A}^{-1}\vec{\omega}_{A}\right)\,.$ (28)

In cases where a prior on the coefficients needs to be imposed, for example
when the magnitude-squared of couplings cannot go negative, as when matching
to specific UV models in Section 7, or when including quadratic dependences on
the coefficients, this analytic method may no longer be used. A numerical MCMC
method using MultiNest has therefore also been implemented in Fitmaker, as
described in Appendix B.

##  6 Global results

###  6.1 Higgs, Diboson and Electroweak fit

The main emphasis of this Section is on the improvements in the Higgs data
since the Run 1 and early Run 2 data that were analysed in the SMEFT framework
in Ellis:2018gqa . However, as has been emphasised previously, e.g., in
Ellis:2014dva ; Ellis:2014jta , there is considerable overlap between the sets
of operators whose coefficients are constrained by both electroweak and Higgs
data 111111We show later in Fig. 11 the effect on the marginalised fit of
removing the LEP EWPO and $WW$ datasets., as visualised in Fig. 1. Therefore
we present in this Section results from a joint fit to the combined Higgs,
diboson and electroweak data, including the 20 operators of relevance listed
in Eq. 8. We recall that the latter are dominated by data from LEP, with the
most important LHC contribution coming from an ATLAS measurement of $M_{W}$.

As already mentioned, this joint fit is carried out to linear order in the
dimension-6 SMEFT operator coefficients, neglecting quadratic dimension-6
contributions to the LHC measurements and linear dimension-8 contributions,
which, as discussed above, are a priori of similar order in the scale
$\Lambda$ of high-mass BSM physics. Details of the fit procedure are described
for the analytic method in Section 5 and a numerical MCMC method in Appendix
B. We use mainly the former, but have verified in representative cases that
fit results do not depend significantly on the method used. Appendix B also
discusses the importance of effects that are quadratic in the dimension-6
operators, and we refer the interested reader to Hays:2018zze  for a
discussion of possible dimension-8 effects in Higgs measurements. We emphasise
again the importance of using, as well as total Higgs production and decay
rates, kinematic measurements of Higgs production as encapsulated in STXS
measurements, due to the different $p_{T}$ dependences of dimension-6
contributions to production amplitudes, whose relative importances are
generally enhanced at higher $p_{T}$ Ellis:2018gqa , as seen in Fig. 2.

Figure 3: Constraints on the indicated individual and marginalised operator
coefficients $C_{i}(1~{}{\rm TeV})^{2}/\Lambda^{2}$ (top and third panels) and
the corresponding scales $\Lambda$ for the indicated values of the ${C_{i}}$
at the 95% confidence level (second and bottom panels), from a combined linear
fit to the Higgs, diboson and electroweak precision observables. In the top
two panels, the bars show the 95% CL ranges from the LHC Run 1 and early Run 2
data (light blue), current data without using the STXS measurements
(intermediate blue), and current data including STXS using either the on-shell
vector boson approximation or the full $1\to 4$ matrix elements for the
4-fermion Higgs decay modes taken from Ref. Brivio:2019myy  (dark blue). In
the bottom two panels, the corresponding marginalised results are indicated by
yellow, orange and red bars, respectively. We also show in purple in the
individual case (grey in the marginalised case) the effect of dropping the
ATLAS $Z+$ jets measurement.

The top panel in Fig. 3 shows the 95% CL intervals for the coefficients
$C_{i}(1~{}{\rm TeV})^{2}/\Lambda^{2}$ of the 20 dimension-6 SMEFT operators
contributing to the joint Higgs, diboson and precision electroweak data
analysis when each operator is analysed individually. We note that certain
coefficients have been scaled as indicated by the labels on the x-axis. In
particular, the Yukawa operator $C_{tH}$, the triple-gauge boson operators
$C_{W}$ and $C_{G}$, and the Higgs-only operator $C_{H\Box}$ have all been
rescaled by 2 orders of magnitude to appear on the same scale. Dotted grey
vertical lines separate the sets of operators that contribute mostly to
electroweak precision observables (EWPO), that contribute mostly to Higgs and
diboson measurements (Bosonic), and that modify the Yukawa couplings (Yukawa).
These categories are indicated as guides to aid the reader; however, as
discussed in more detail later, there are correlations between these sectors.

The two right-most bars for each operator are from fits using all the
available Higgs and diboson data from Run 2 of the LHC together with the
precision electroweak constraints, differing only in whether they include the
STXS constraints. The intermediate blue bars demonstrate the effect of
replacing the combined ATLAS STXS dataset with the latest combined ATLAS
signal strengths, in which case there are significantly weaker constraints for
some of the operator coefficients. These results do not differ significantly
whether predictions in the on-shell approximation are used, where the $H\to
4f$ process is assumed to originate from an underlying $H\to VV^{*}\to 4f$
decay with one gauge boson taken to be on-shell, or whether the SMEFT
dependence of the full $H\to 4f$ matrix elements is used (see Brivio:2019myy
and Section 5). For comparison, the left-most bar (light blue) is from a fit
in which only the Higgs and diboson data analysed in Ellis:2018gqa  are used.
In the case of $C_{W}$, we also show (in purple) the effect of dropping the
ATLAS $Zjj$ measurement Aad:2020sle , which does not impact significantly the
constraints on the other operator coefficients.

In general, the individual fit ranges for the first 10 operator coefficients
of Fig. 3 counting from the left, i.e., up to and including $C_{Hu}$, are very
similar in fits including all the LHC Run 2 data to those found using the
earlier set of Higgs, diboson and electroweak data. This is because the
precision electroweak data provide the strongest constraints on these
individual operator coefficients. The impacts of the Higgs and diboson data
are more apparent for the rest of the operator coefficients, i.e., from
$C_{H\Box}$ rightwards, particularly for the Yukawa operators that have
benefited from improved sensitivity of the Higgs couplings to the tau and
bottom.

Using the same colours, the bars in the second panel show the 95% CL lower
limits on the $\Lambda_{i}$ on logarithmic scales in units of TeV, for
different values of ${C_{i}}$. These reaches are estimated by taking half the
width of the 95% CL ranges of $C_{i}/\Lambda_{i}^{2}$ as the typical
sensitivity of the measurement. Here and in subsequent analogous panels, the
darker (lighter) coloured histograms are for $C_{i}=0.01(1)$ and the
histograms with dashed outlines are for the strong-coupling perturbativity
limit $C_{i}=(4\pi)^{2}$. In general, the $\Lambda_{i}$ scales would be
modified by a factor $\sqrt{C_{i}}$, which would depend on whether the Wilson
coefficient is induced by strongly- or weakly-coupled new physics, at tree- or
loop-level.

The corresponding 95% CL constraints for the marginalised case, where we
include simultaneously all operators in the analysis and then profile the
likelihood over all coefficients except one, as described in Section 5, are
shown in the lower pair of panels in Fig. 3. The yellow, orange and red bars
from left to right are the fits to the old data, the new data without the STXS
measurements, and including them, respectively. There is again no significant
difference for different treatments of $h\to 4f$. The marginalised constraints
are weaker overall than the individual constraints, as the fit is allowed to
explore all possible variations in the space of coefficients. We also note
that the STXS measurements play a key role in the marginalised constraints for
some operators, e.g., $C_{HG},C_{G}$ and $C_{tH}$. As discussed in Section
4.2, removing them causes a degeneracy in the parameter space that prevents
meaningful constraints in these directions. We show in dark brown the impact
on $C_{W}$ of dropping the $Zjj$ constraint, which still does not
significantly affect the other operators. The more traditional diboson
measurements suffer from suppressed SM interference at high energy due to
helicity selection Azatov:2016sqh . This is why the bound without $Zjj$ is
dominated by the LEP 2 data, with the high mass $W^{+}W^{-}$ distributions
from the LHC yielding no significant improvement. The $Zjj$ observable is
therefore extremely useful for constraining anomalous gauge boson self-
interactions at linear level, overcoming the non-interference issue and
accessing the leading contributions in the SMEFT expansion.

We see that most of the 95% CL ranges are reduced when the full Run 2 data are
included, some quite substantially, the only exceptions being $C_{G}$ and
$C_{HG}$. This occurs despite the individual constraints improving in both
parameter space directions. We attribute the slight worsening of the
marginalised bounds to the presence of a highly boosted $H\to b\bar{b}$
measurement in the 2018 data Sirunyan:2017dgc  that selects Higgs $p_{T}$ >
450 GeV, which is significantly higher than the highest, $p_{T}$ > 200 GeV of
the stage 1.0 STXS bins used in our 2020 dataset. Removing this observable
degrades the 2018 bounds below our most recent results. We expect this
sensitivity to be recovered once the stage 1.2 measurements, which probe a
similarly high $p_{T}$ region, are incorporated. We find $\chi^{2}/{\rm
dof}=0.90$ ($p=0.82$) for our flavour-universal global fit, to be compared
with $\chi^{2}/{\rm dof}=0.88$ ($p=0.84$) for the SM. Among the 20 operators
considered in the fit to the Higgs, diboson and electroweak data, the most
weakly constrained operator coefficients are all constrained so that
$\Lambda/\sqrt{C_{i}}\gtrsim 500\,(400)$ GeV in the individual (marginalised)
fits, suggesting that the linear SMEFT treatment may be adequate in this
sector Contino:2016jqw .

###  6.2 Top fit

Figure 4:  Left panel: Breakdown of the impacts of various top datasets on the
2-dimensional subspace of the four-fermion operators
($C_{Qq}^{1,8},C_{tq}^{8}$), setting all other operator coefficients to zero.
Right panel: Similar breakdown of the impacts of various top datasets on the
2-dimensional subspace of the four-fermion operators
($C_{Qu}^{8},C_{Qd}^{8}$), setting all other operator coefficients to zero.

Figure 5:  Constraints on the indicated individual and marginalised operator
coefficients at the 95% confidence level (upper and lower figures,
respectively), from a combined linear fit to the top data and electroweak
precision observables. The impact of $t\bar{t}$ data is highlighted by the
evolution of the constraints starting from no $t\bar{t}$ data (light
blue/yellow) adding Run 1 $t\bar{t}$ total and differential cross-section data
(blue/pink), the corresponding Run 2 $t\bar{t}$ data (purple/orange), and
finally $t\bar{t}$ asymmetry measurements $A_{FB}$ from the Tevatron and
$A_{C}$ from the LHC (green/red).

We consider now the information provided by top quark data, in a dedicated fit
to a subset of relevant ‘top quark operators’, i.e., those involving at least
one top quark field (the top Yukawa operator, and the top 2F and top 4F
categories in Equation 2.2.2) plus the triple gluon operator, $C_{G}$, which
globally affects most QCD-induced processes. We also include EW precision
observables, which close a single blind direction
($C_{HQ}^{(1)}+C_{HQ}^{(3)}$) affecting the left-handed $Zb\bar{b}$ coupling,
as well as the latest $t\bar{t}H$ signal strength measurements from the two
CMS and ATLAS Higgs combination papers included in the final fit, to constrain
the top Yukawa operator. Overall, the $\chi^{2}/n_{\text{dof}}$ for the SM is
0.95. However, there are a number of observables that exhibit some tension
with the SM predictions, such as the recent 13 TeV measurement of the
differential $p_{T}$ distribution in $t$-channel single top production
Sirunyan:2019hqb  ($\chi^{2}/n_{\text{d.o.f}}=5.3$), $t\bar{t}W$ cross section
measurements by CMS at 8 TeV Khachatryan:2016fky  and ATLAS at 8 and 13 TeV
Aad:2015eua ; Aaboud:2019njj  ($\chi^{2}/n_{\text{d.o.f}}=2.6,2.1$ and 1.9),
and CMS $t\bar{t}$ differential distributions at 8 TeV
($\tfrac{d\sigma}{dm_{t\bar{t}}y_{t\bar{t}}}$ in the l+jets channel) and 13
TeV ($\tfrac{d\sigma}{dm_{t\bar{t}}}$ in the dilepton channel), both with
$\chi^{2}/n_{\text{d.o.f}}=1.6$. These tensions may lead to a preference for
non-zero Wilson coefficients, though this depends on whether other, more
consistent observables also constrain the operators in question.

Fig. 5 shows the 95% CL intervals for the coefficients individually (top two
panels) and after marginalisation (bottom two panels). The pale blue bars in
the top two panels are obtained without using any $t\bar{t}$ data, the
intermediate blue bars are obtained including the Run 1 $t\bar{t}$ total and
differential cross-section data, the dark blue bars include also the
corresponding Run 2 $t\bar{t}$ data, and finally the green bars include also
the $t\bar{t}$ asymmetry measurements $A_{FB}$ from the Tevatron and $A_{C}$
from the LHC. When removing the $t\bar{t}$ data entirely, a closed fit is not
possible, so marginalised constraints do not exist. The corresponding colours
for the latter three sets of data (Run 1 $t\bar{t}$, including Run 2, and
including $A_{FB}$ asymmetries) in the bottom two panels are yellow, orange
and red. We also show in dark brown the impact of removing electroweak
precision observables for the two operators most affected, $C_{HQ}^{(1)}$ and
$C_{Ht}$. As previously, the constraints on the scales $\Lambda/\sqrt{C_{i}}$
are estimated by taking half the width of the 95% CL ranges of
$C_{i}/\Lambda_{i}^{2}$ for $C=1$ as the typical sensitivity of the
measurement. We note that many of the constraints on the quantities
$C_{i}(1{\rm TeV})^{2}/\Lambda^{2}$ in the individual and marginalised cases
differ by an order of magnitude, as shown in the top and third panels of Fig.
5, and that the corresponding constraints on the scales $\Lambda/\sqrt{C_{i}}$
are typically a factor $\sim 3$ stronger in the individual analysis, as seen
in the second and bottom panels.

The impact of the tension with the SM for the aforementioned observables can
be seen from the individual constraints of Fig. 5. We see that $C_{tH}$ and
the ‘top 2F’ category appear consistent with the SM. Although it disagrees the
most with the SM, the single top differential $p_{T}$ data does not lead to
significant pulls for the operators that can affect it, $C_{Qq}^{3,1}$,
$C_{HQ}^{3}$ and $C_{tW}$. This is due to the fact that single operators are
not able to improve the fit significantly, given that the disagreement
predominantly comes from the lowest $p_{T}$ bin, combined with the relatively
good agreement of most other single-top data. Instead, the effect of the
$t\bar{t}W$ data can be seen in the individual constraints without $t\bar{t}$
data, that cause the deviation of the best-fit values for $C_{Qq}^{1,8}$ and
$C_{Qq}^{3,8}$ away from zero in the positive and negative directions,
respectively. This is consistent with a relative minus sign in the linear
dependence of the $t\bar{t}W$ process on these two operators. On the other
hand, $t\bar{t}$ data depend on 9 of the operators in question, namely all of
the ‘top 4F’ category except $C_{Qq}^{3,1}$, as well as $C_{tG}$ and $C_{G}$.
Gradually adding the differential $t\bar{t}$ data draws the coefficients
towards negative values, resulting in particularly large pulls, especially for
$C_{G}$. Finally, the $t\bar{t}$ asymmetry observables add an orthogonal
constraint on the four-fermion operators that restore consistency with the SM
in this sector. Since $C_{G}$ does not produce an angular asymmetry in the
$t\bar{t}$ matrix element, its significant, non-zero best-fit value remains.
Considering now the marginalised results, we see that more $t\bar{t}$ data and
the inclusion of asymmetries has a significant impact on the global
sensitivity, even indirectly affecting sensitivity to EW top quark couplings
$C_{Ht}$ and $C_{tB}$ by constraining the allowed four-fermion contributions
to $t\bar{t}Z$. We also show the importance of EW precision observables in
closing the parameter space for neutral top quark couplings, by noting the
large significant degradation of the limits in this space when removing them
from the fit. All of the potentially large pulls are washed out by the
marginalisation, except for $C_{G}$, which lies 4.2 standard deviations away
from zero.

The interplay between different $t\bar{t}$ measurements in constraining the
four fermion sector is shown in Fig. 4, inspired by a similar analysis in Ref.
Brivio:2019ius . Two pairs of operators are selected, setting all other
coefficients to zero: $C_{Qq}^{1,8}$ and $C_{tq}^{8}$, which couple to left-
and right-handed top quarks, respectively, in the left panel, and
$C_{Qu}^{1,8}$ and $C_{Qd}^{8}$, which couple left handed top quarks to the up
and down quarks, respectively, in the right panel. We see explicitly the
complementarity between the $t\bar{t}$ cross-section measurements, which
strongly constrain one linear combination of $C_{Qq}^{1,8}$ and $C_{tq}^{8}$,
and the asymmetry measurements, which constrain an orthogonal direction. The
combination of these measurements constrains each of $C_{Qq}^{1,8}$ and
$C_{tq}^{8}$ quite tightly, though less so in the marginalised case, indicated
by the dashed lines. The fact that forward-backward asymmetries are sensitive
to the chiral structure of the $t\bar{t}$ matrix element explains why they
excel at distinguishing modified interactions for left- and right-handed tops
but less so the isospin of the initial state quark ($u$ or $d$). In this case
the combination of $t\bar{t}$ measurements constrains a highly correlated
combination of the coefficients, though each is only weakly constrained in the
marginalised fit, as seen in the bottom two panels of Fig. 5. The large
differences between their individual and marginalised limits that we observe
indicate strong correlations among the top quark four fermion operators
constraints.

Overall, the best-constrained coefficients in Fig. 5 are $C_{tG}$,
$C_{HQ}^{(3)}$, $C_{HQ}^{(1)}$, $C_{Qq}^{3,1}$ and $C_{G}$, with one direction
being driven by electroweak precision observables. The large negative values
of $C_{G}$ can be traced back to $t\bar{t}$ differential cross-section
measurements, and we discuss a specific example in Section 6.3.3 below, where
we also consider the possibility that $C_{G}$ is very small, as suggested on
the basis of a quadratic analysis of multijet data. Some less well-constrained
operator coefficients are $C_{8}^{Qd},C_{8}^{tu}$ and $C_{8}^{td}$, whose
scales $\Lambda$ are $\gtrsim 800$ GeV in the individual analysis and $\gtrsim
250$ GeV in the marginalised analysis when the corresponding $C_{i}=1$, in
which case the validity of the global SMEFT interpretation for these operators
could be questioned for weakly-coupled UV completions, given that some of the
$t\bar{t}$ data extends up to TeV energies. The operator coefficients
$C_{Ht},C_{tB}$ are the least constrained, as neutral top quark interactions
are particularly hard to constrain. Production of $t\bar{t}Z$ and $tZ$ are the
main handles we have on these couplings, and these are still not so well
measured and lacking in differential data. Unfortunately, $C_{Ht}$ does not
predict any effects that grow in energy in either of these processes, and
$C_{tB}$ has a suppressed interference with the SM, meaning that one does not
expect significant gains from differential measurements, especially in a
linear analysis Bylund:2016phk ; Degrande:2018fog . Other rare EW top
processes, such as $t\bar{t}Wj$ and $tWZ$ have been shown to be sensitive to
such unitarity-violating behaviour and will therefore provide useful
constraining power, once they are measured at the LHC Dror:2015nkp ;
Maltoni:2019aot . However, in all cases we see that the dashed histograms
extend beyond a TeV even in the marginalised case , indicating that the SMEFT
analysis should be a good approximation in the strong-coupling limit
$C_{i}=(4\pi)^{2}$.

Figure 6:  Results from the global fit to the electroweak, diboson, Higgs and
top data in the top-specific SU(2)${}^{2}\times$SU(3)3 scenario. Top two
panels: fits to the individual operators, showing the 95% CL ranges for the
operator coefficients $C_{i}$ normalising the new physics scale $\Lambda$ to 1
TeV, and the ranges for the scales $\Lambda$ for different values of the
$C_{i}$. Bottom two panels: similar, but marginalising over the other
operators. In the top panels, fit results found using only electroweak,
diboson and Higgs data are shown in light blue, those found in the combination
with the top data are shown in dark blue, and those using only top data are
shown in green. In the bottom panels, the corresponding marginalised fit
results are shown in yellow, red and pink.

###  6.3 Combined Top, Higgs, Diboson and Electroweak Fit

As we discussed in Section 2, there are different possible treatments of the
flavour degrees of freedom within the SMEFT framework. Specifically, in this
paper we assume either an SU(3)5 symmetry in the operator coefficients (broken
by the Yukawa operators) or allow this symmetry to be broken to an
SU(2)${}^{2}\times$SU(3)3 symmetry by the coefficients of third-generation
fermions in a top-specific flavour scenario.

We find that, among the 20 flavour-universal operators in the SU(3)5 scenario,
only the marginalised constraints on $C_{G},C_{HG}$ and $C_{tH}$ are improved
significantly by the additional top measurements beyond the constraints that
are already provided by the electroweak, diboson and Higgs data, while the
sensitivity to $C_{Hq}^{(1)}$ decreases. This indicates the robustness of the
fit despite the increase in the number of parameters. However, there are
significant correlations between the datasets. Accordingly, in the following
we focus attention on the results from a simultaneous global fit to the 34
operators of the top-specific flavour scenario that was described in Section
2.2.2.

####  6.3.1 Operator sensitivities in individual and marginalised fits

Figs. 6 shows the results from this combined fit to all the available Higgs,
electroweak, diboson and top data, switching on one operator at a time (top
two panels) and marginalising over all other coefficients (bottom two panels),
respectively. In each case, the upper panel shows the 95% CL ranges for the
operator coefficients $C_{i}$ normalising the corresponding new physics scales
to 1 TeV. As indicated on the x-axis labels, certain operator coefficients
have been rescaled for the sake of convenience. The bars in the lower panels
show the 95% CL lower limits on the $\Lambda_{i}$ on logarithmic scales in
units of TeV. As previously, these reaches are estimated by taking half the
width of the 95% CL ranges of $C_{i}/\Lambda_{i}^{2}$ for $C=1$ as the typical
sensitivity of the measurement.

The differences in the constraints on the 20 operators entering the flavour-
independent SU(3)5 fit between including top data or not are small in the
individual case (dark vs light blue), except for $C_{G}$. Marginalising widens
the ranges allowed by the fit, but the effect of marginalising over a larger
set of coefficients - 34 compared to 20 in the top-specific flavour scenario
compared to the flavour-universal case (red vs yellow), introduces noticeable
differences for only a few operator coefficients, namely $C_{G},C_{HG}$ and
$C_{tH}$. The differences in the constraints on the top operators (shown in
green in the individual case and pink in the marginalised case) when the
electroweak, diboson and Higgs data are included in the fit are generally
small, apart in the case of the Yukawa operator, $C_{tH}$. The loop-level
constraints from Higgs production via gluon fusion are clearly more powerful
in constraining this operator individually, but this sensitivity is diluted by
marginalisation, which allows the other operators affecting this process to
float.

Overall, the data are sufficient for a closed fit with no flat directions, and
we find $\chi^{2}/{\rm dof}=0.79$ ($p=1.0$) for our top-specific global fit.
The sensitivities to the scale of new physics in the operator coefficients in
the individual case are generally several hundred GeV or more for $C_{i}=1$,
the exception being the operator $C_{tB}$, which is essentially decoupled from
the other operators, as we discuss in more detail later. We also note that the
$C_{i}$ for the Yukawa operators would be expected to contain a Yukawa factor
in the MFV hypothesis, as discussed in Sec. 2, with the bounds then weakened
appropriately. The scale sensitivities in the marginalised case still reach a
TeV for $C_{i}=1$ for most of the electroweak precision observables set and
some of the operators in the bosonic and top categories, falling to $\sim 300$
GeV for some of the top operators. The weakest scale constraint is again that
for $C_{tB}$, reaching only $\sim 100$ GeV. We emphasise, however, that
specific UV completions each generate only a subset of operators, for which
they have correspondingly improved reaches. The individual and marginalised
fits can therefore be taken as optimistic and pessimistic sensitivity
estimates, respectively, with realistic cases intermediate between the two.

####  6.3.2 Sensitivities in ‘Higgs-only’ operator planes

Figure 7:  Constraints on the indicated pairs of operator coefficients at the
95% confidence level, marginalised over the remaining degrees of freedom in
the ‘Higgs only’ operator set. The green and mauve shaded areas correspond to
combined linear fits to Higgs data and Higgs + top data, respectively. The
blue ellipses indicate the marginalised constraints from Higgs + top data
after introducing top-quark four-fermion operators into the fit, and the
yellow ellipses are obtained from a fit dropping the $t\bar{t}H$ data. Figure
8:  Constraints on the indicated pairs of operator coefficients at the 95%
confidence level, setting the other operator coefficients to zero. The shaded
regions correspond to linear fits to Higgs signal strengths and 0 jet STXS
bins (blue), $t\bar{t}H$ signal strengths (mauve), $\geq 1$ jet STXS bins
(orange) $t\bar{t}$ data (green), $t\bar{t}V$ data (red) and their combination
(grey). The dashed ellipses show the constraints obtained by marginalising
over the remaining Wilson coefficients of the full fit.

In order to assess the potential impact of the interplay between top and Higgs
data, we may consider the following subset of ‘Higgs-only’ operators:

$\\{C_{H\Box},C_{HG},C_{HW},C_{HB},C_{tH},C_{bH},C_{\tau H},C_{\mu H}\\}$ (29)

together with $C_{G}$ and $C_{tG}$, which do not modify Higgs interactions
directly but can impact gluon fusion. Performing a fit to this subset, Fig. 7
displays the result for the 95% CL constraints when top data are combined with
Higgs data in planes showing different pairs of the operator coefficients
$C_{HG},C_{tG},C_{tH}$ and $C_{G}$, marginalised over the other coefficients
in (29). This is the relevant set of operators in which the interplay between
Higgs and top physics is most evident, taking place in the gluon fusion and
$t\bar{t}$ associated Higgs production processes. It is well known that there
is a degeneracy in gluon fusion between $C_{HG}$ and $C_{tH}$ that prevents it
from being used as a robust indirect constraint on the top Yukawa coupling, or
conversely, heavy coloured particles that couple to the Higgs. This strongly
motivates the direct measurement of the top Yukawa, via $t\bar{t}H$. In Fig.
7, the yellow ellipses show results for Higgs data, without $t\bar{t}H$, while
the green ellipses show the sensitivity for Higgs data including it,
indicating how a relative flat direction in this plane (top left panel) is
lifted by the inclusion of $t\bar{t}H$. However, despite not being directly
related to Higgs couplings, both $C_{tG}$ and $C_{G}$ can also affect
Higgs(+jet) production in gluon fusion. Thankfully, these can be constrained
by top data, particularly $t\bar{t}$ (and multi-jet data at order
$\Lambda^{-4}$, for the latter). The results for the combination of Higgs with
top data are shown as mauve ellipses, where we see in each plane a very
substantial reduction of the area allowed at the 95% CL. The difference
between the two sensitivities underlines the strength and importance of this
data in indirectly pinning down BSM interactions of the Higgs, where now the
$C_{tG}$ and $C_{G}$ directions are squeezed down by an order of magnitude. We
also see that several (anti)correlations between pairs of operator
coefficients are suppressed when top data are included, most noticeably in the
$(C_{HG},C_{G})$ plane. However, using top data to constrain only two
operators is not in keeping with the global spirit of SMEFT interpretations,
especially given the large number of degrees of freedom discussed in Sec.
2.2.2 that could potentially dilute its power to bound $C_{tG}$ and $C_{G}$.
We address this question by increasing our operator subset to include the 7
four-fermion operators that impact $t\bar{t}$ production, with the new
marginalised constraints shown as blue ellipses. Surprisingly, there is little
further change when adding the four-fermion operators, indicating a very
limited dilution effect and underlining the robustness of the complementarity
of top data in indirectly constraining Higgs couplings.

Fig. 8 displays the constraints on the same pairs of operator coefficients at
the 95% confidence level when the coefficients of other operators are set to
zero, with more fine-grained information on the constraints provided by the
different datasets. The shaded regions are the results of linear fits to Higgs
signal strengths and 0 jet STXS bins (blue), $t\bar{t}H$ signal strengths
(mauve), $\geq 1$ jet STXS bins (orange), $t\bar{t}$ data (green), $t\bar{t}V$
data (red) and their combination (grey). The dashed ellipses show the
constraints in the corresponding parameter planes when marginalising over the
remaining Wilson coefficients of the full fit, as shown in Fig. 7. As was to
be expected, the constraints obtained when the other operator coefficients are
set to zero are significantly stronger. The complementarity between $ggF$,
$t\bar{t}H$ and $t\bar{t}$ is again evident, with $t\bar{t}V$ data also
providing some additional information on $C_{G}$. We also see that $\geq 1$
jet STXS bins have not yet reached the level of precision needed to offer
significant complementary information in this parameter space.

####  6.3.3 The triple-gluon operator $C_{G}$

The operator $\mathcal{O}_{G}$ consists of triple-gluon field strengths
Simmons:1990dh ; Dreiner:1991xi ; Cho:1994yu  and so affects any observables
sensitive to jets Ghosh:2014wxa ; Hirschi:2018etq ; Goldouzian:2020wdq ;
Krauss:2016ely . This includes many of the Higgs and top processes in our
global fit, as shown in Fig. 12 that we discuss in the next Section, where we
see sizeable correlations of $C_{G}$ with the operators
$C_{HG},C_{tH},C_{tG},C_{Qq}^{3,8},C_{Qq}^{1,8},C_{Qu}^{8},C_{tq}^{8},C_{tu}^{8}$
and $C_{td}^{8}$, spanning both the Higgs and top sectors. Therefore, the
gluonic operator complicates the SMEFT interpretation of the measurements in
these sectors.

Figure 9:  The measurement of the $t\bar{t}$ invariant mass distribution in
the lepton+jets channel at 13 TeV by the CMS experiment Sirunyan:2018wem
compared to to the SM prediction at the NNLO QCD + NLO EW level Czakon:2017wor
. Also shown are predictions corresponding to the best-fit values for
$\hat{C}_{G}$ (green upward triangles), $\hat{C}_{tG}$ (red downward
triangles), their combination (purple circles) and the global best-fit point
in the full parameter space (orange stars).

It has been argued in Refs. Krauss:2016ely ; Hirschi:2018etq ;
Goldouzian:2020wdq  that a very strong constraint on $C_{G}$ is provided by
multijet data that are not included in our default data set, and that one can
set $C_{G}=0$ when analysing electroweak, Higgs or top data. However, this
strong constraint relies on quadratic contributions to multijet observables,
whereas our global fit is made to linear order. The linear contributions of
$C_{G}$ are small since the amplitude involving $C_{G}$ in gluon-gluon
scattering does not interfere with the SM, and in quark-gluon scattering it
does so only proportionally to the quark masses. This has made it a challenge
to constrain in past studies Simmons:1990dh ; Dreiner:1991xi ; Cho:1994yu .
However, the top sector with its large quark mass provides an opportunity to
recover sensitivity at linear order to $C_{G}$, as studied most recently in
Ref. Bardhan:2020vcl . Here we see, comparing Figs. 3, 5 and 6 that the top
data indeed provide the strongest constraints on $C_{G}$ and, moreover, these
bounds are robust when marginalising over the other operators in both the
Higgs and top sectors.

The $C_{G}$ fit also shows the strongest pull away from zero, with a
significance $\sim 4.3\,\sigma$ towards negative values for the individual
fit, and $C_{G}$ is $4.1\,\sigma$ away from zero at the best fit in the
marginalised case, as shown in Fig. 6. This effect is due to $t\bar{t}$
differential data, an example of which is given by the 13-TeV invariant mass
distribution data shown in Fig. 9. We see there that the $m_{t\bar{t}}$
dependence of the cross-section data, normalised to the SM prediction, denoted
by black points, differs quite significantly from that of the SM, represented
by grey boxes at a value of 1, and it is this discrepancy that pulls $C_{G}$
into negative territory, as shown by the green triangles, plotting the best-
fit $C_{G}$ contribution. We see that the agreement with the data is obtained
by an interplay between $C_{G}$ and $C_{tG}$, whose best fit prediction, shown
by the red triangles, improves the agreement in the low mass bins while not
too significantly affecting the high mass region. The sum of the best-fit
$C_{G}$ and $C_{tG}$ predictions, shown by the purple circles, coincide with
the global best-fit prediction, shown by the orange stars, demonstrating that
the fit to the data is obtained primarily by this interplay. We emphasise,
however, that the significance of this effect could be reduced if there were
some important contribution to the $m_{t\bar{t}}$ distribution close to
threshold that has not been included in the SM calculation.

Figure 10:  Same as Fig 7, but setting $C_{G}=0$.

Accordingly, we have investigated the consequences of assuming that $C_{G}$
can be better constrained by including dedicated QCD analyses, and have
analysed the effects of setting $C_{G}=0$ in our study of the impact of top
data on Higgs coupling measurements, as well as in our marginalised global
fits in the flavour-universal SU(3)5 and the top-specific
SU(2)${}^{2}\times$SU(3)3 scenarios. Fig. 10 shows that the global space of
constraints is changed compared to Fig. 7, especially when only including
Higgs data (green ellipse), with improved sensitivity (see Fig 11) and a
different pattern of correlations. Once the top data is included, the overall
sensitivity is improved, and the best-fit point before including the four-
fermion operators (mauve ellipse) is closer to the SM. The relative impact of
adding the four-fermion operators (blue ellipse) is more noticeable when
$C_{G}=0$ than when it is non-zero. When $C_{G}=0$, the tension with the SM in
the $t\bar{t}$ data pulls the four-fermion operators and $C_{tG}$. These
operators also affect $t\bar{t}H$, and $C_{tG}$ also modifies gluon fusion,
leading to a cascade of shifts in the ranges of these operators. Moving to the
global results, we see in the top two panels of Fig. 11 that this constraint
has little effect on the 95% ranges we find in our marginalised SU(3)5 fit,
except that the ranges of $C_{HG}$ and $C_{tH}$ are reduced noticeably when we
set $C_{G}=0$ 121212For completeness, we also show in the top two panels of
Fig. 11 the effects of dropping the EWPOs (and LEP $WW$) from the marginalised
SU(3)5 fit (yellow bars). As could be expected, there are large effects on the
constraints on the operators that contribute most to the EWPOs, but quite
small effects in the bosonic and Yukawa sectors., as expected from the
previous discussion of Fig. 10. In the SU(2)${}^{2}\times$SU(3)3 (bottom two
panels of Fig. 11) there are shifts in the central values of several top
operator coefficients, with the four fermion operators moving further away
from the SM to absorb the aforementioned discrepancies in top data. Overall,
no significant reductions in the ranges of any operator coefficients are
observed, and thus our fit results are relatively insensitive to the treatment
of $C_{G}$.

Figure 11: Comparison of the constraints on the indicated marginalised
operator coefficients $C_{i}(1~{}{\rm TeV})^{2}/\Lambda^{2}$ (top and third
panels) and the corresponding scales $\Lambda$ for the indicated values of the
${C_{i}}$ at the 95% confidence level (second and bottom panels), found in a
combined linear fit to the Higgs, diboson and electroweak precision
observables (top two panels) and including in addition top data (bottom two
panels), including $C_{G}$ in the fit (orange) and setting $C_{G}=0$ (red).
Also displayed in yellow in the top two panels is a fit without LEP (EWPO and
$WW$) measurements.

####  6.3.4 Correlation matrix and principal component analysis

The full correlation matrix for the top-specific marginalised fit is shown in
Fig. 12, colour-coded and labelled in percentages. Some of these correlations
can be explained intuitively by the simple fact that two operators contribute
to the same observable, while others occur more indirectly, though a chain of
dependencies that is difficult to trace through the inversion of the Fisher
information matrix. We see that there are substantial (anti) correlations
between the coefficients of operators in the EWPO and Bosonic sectors, and
there are also many (anti)correlations within the top 4F sector. On the other
hand, there is only one large off-diagonal entry in the top 2F sector, namely
a negative correlation between $C^{(3)}_{HQ}$ and $C^{(1)}_{HQ}$. Along with
$C_{Ht}$, that is mildly correlated with these two, these operators contribute
to neutral top quark couplings. As previously discussed, one linear
combination, $C^{(3)}_{HQ}+C^{(1)}_{HQ}$ modifies the $Zb\bar{b}$ coupling,
and hence has a very strong LEP constraint, while the other two can only be
probed in EW top processes. The bottom Yukawa operator ($C_{bH}$) exhibits
some correlations with those of the top ($C_{tH}$) and the tau ($C_{\tau H}$),
and in the bosonic sector some moderate correlations are observed, notably
between $C_{HW}$, $C_{HB}$ and $C_{H\Box}$, and between $C_{HG}$ and $C_{G}$
(as expected from the discussion in Sec. 6.3.2). Turning to correlations
between operators in different sectors, we note substantial positive
correlations between $C_{bH}$ and both $C_{Hl}^{(3)}$ and $C_{Hq}^{(3)}$, in
the Yukawa and EWPO sectors, respectively, as well as a substantial negative
(positive) correlation between $C_{tH}$ in the Yukawa sector and
$C_{H\Box},C_{G}$ ($C_{HG}$) in the Bosonic sector. Finally, there are several
large (anti)correlations between operators in the top 4F and top 2F sectors,
namely $C^{3,1}_{Qq}$ and $C_{HQ}^{(3)}$ (positive), $C^{3,1}_{Qq}$ and
$C_{HQ}^{(1)}$ (negative), $C^{8}_{Qu}$ and $C_{Ht}$ (negative), and
$C^{8}_{tq}$ and $C_{Ht}$ (positive). Overall, there are 18 correlation
coefficients with magnitude $\geq 0.2$ between operators in a top sector on
the one hand and in a Yukawa, bosonic or electroweak sector on the other hand.
These and the top sector may not be talking to each other very loudly, but
they are starting to whisper to each other.

Figure 12: The full $34\times 34$ correlation matrix for the marginalised top-
specific fit. The operators are grouped into those affecting primarily
electroweak precision observables, bosonic observables, Yukawa measurements
and top electroweak measurements, as well as top-quark four-fermion operators.
The entries in the correlation matrix are colour-coded according to the
indicated magnitudes of the correlation coefficients.

In the Gaussian approximation to the global likelihood that is used here, it
is also informative to diagonalise the constraints on the operator
coefficients in the orthonormal eigenvector basis and perform a principal
component analysis. This tells us which directions are most constrained in the
fit, and what operators contribute to those directions. We display the
constraints on the eigenvectors graphically in Fig. 13: the rows in the centre
panel correspond to the different operator coefficients, the columns
correspond to the different eigenvectors, and the colour-coded squares
represent the moduli-squared of the operator components in the eigenvectors.
The latter are ordered such that the strengths of the global constraints
decrease from left to right, as seen in the top panel of Fig. 13, where the
95% CL bounds on the scales $\Lambda$ are calculated assuming that the linear
combination of operator coefficients making up that particular eigenvector is
set to unity. The bottom panel tabulates the respective relative constraining
powers of the electroweak precision data, LEP diboson data, Higgs coupling
strength measurements from Runs 1 and 2, STXS measurements, LHC diboson and
$Zjj$ measurements, $t\bar{t}$ measurements, single top measurements and
$t\bar{t}V$ measurements. These are defined as the relative contribution of
each dataset to the corresponding entry of the diagonalised Fisher
information. Entries where there is no significant constraining power are
indicated by “-”.

We see that the largest component of the best-constrained eigenvector is
${\cal O}_{HB}$, and that the limit on its scale exceeds 20 TeV, with the most
important contribution coming from the STXS measurements, followed by Higgs
signal strength measurements. The scales of three more eigenvectors are
constrained beyond the 10-TeV level, with the most important contributions
coming increasingly from the electroweak precision measurements as well as
STXS and Higgs signal strength measurements. They can broadly be associated
with the powerful sensitivity we have obtained in constraining the
$H\gamma\gamma$ and $Hgg$ interactions. The first 11 eigenvectors are
constrained by a mixture of EWPO and Higgs data, showing that these two sets
are providing complementary and competitive bounds in the multi-TeV range.

We next see a particularly strong constraint coming entirely from single top
data on $C_{Qq}^{3,1}$ alone. We note that other operators contribute along
that eigenvector direction, which is given by
$-0.99C_{Qq}^{3,1}+0.06C_{HQ}^{(3)}-0.06C_{HQ}^{(1)}-0.06C_{HW}$, but with too
small a magnitude to be visibly coloured 131313For completeness, the numerical
values of the eigenvector components are provided in Table 7.. However, this
is responsible for the large correlation between $C_{Qq}^{3,1},C_{HQ}^{(3)}$
and $C_{HQ}^{(1)}$ shown in Fig. 12. Several other examples of isolated
operators can be found across the figure, identified by the columns dominated
by a single, very dark spot. Here the eigen-directions nearly coincide with a
particular operator, such that the rest of the fit should be relatively
independent of whether these are included or not. Specific examples are
$C_{\mu H}$, which is constrained in isolation by the $H\to\mu\mu$ signal
strength, and $C_{tW}$, which is constrained mainly by $W$-helicity fraction
measurements in $t\bar{t}$ data. We also see that $Zjj$ mostly constrains
$C_{W}$ with not much effect on the rest of the fit.

The least strongly constrained eigenvector is predominantly ${\cal O}_{tB}$,
whose scale is bounded just above 100 GeV when the operator coefficients are
normalised to unity, mainly by single top and $t\bar{t}$ data. There are three
more eigenvectors with scales $\lesssim 300$ GeV whose principal components
are top operators. The most important constraints on these eigenvectors are in
the top sector, principally from the $t\bar{t}$ cross-section and $t\bar{t}V$
measurements. As discussed in Section 6.2, while the validity of the SMEFT may
be questioned when the operator coefficients are normalised to unity, it
should be reliable for all top operators in the strong-coupling limit. Also,
we expect the SMEFT to be valid for the better-constrained eigenvectors even
for unit-normalised coefficients, since these eigenvectors have relatively
small top operator components, as seen in the middle panel of Fig. 12.

Figure 13:  Constraints on the eigenvectors in the Gaussian approximation to
the global likelihood function. The left panel shows the strengths of the
global constraints on the eigenvectors. The columns in the centre panel
correspond to the different operator coefficients, the rows correspond to the
different eigenvectors, and the colour-coded squares represent the moduli-
squared of the operator components to the eigenvectors. The right panel shows
the constraining powers of the electroweak precision data, LEP diboson data,
Higgs coupling strength measurements from Runs 1 and 2, STXS measurements and
LHC diboson and $Zjj$ measurements, respectively. Instances where there is no
significant constraining power are indicated by “-”.

##  7 Constraints on UV Completions

So far our approach has been to use the SMEFT framework to combine all
relevant data inputs and perform a global fit to the coefficients of the
dimension-6 operators characterising possible modifications of the SM
Lagrangian at leading order in the momentum expansion. The result of this fit,
summarised in Fig. 6, provides a model-independent way to evaluate the
compatibility of the SM with the available experimental data. This SMEFT
analysis provides information on the level of new physics contributions
compatible with the current data. We presented in the bottom panel of Fig. 6
limits on the dimensionful parameters $\Lambda/\sqrt{C_{i}}$, which can be
interpreted as constraints on the possible scale of New Physics compatible
with current measurements. The individual limits correspond to bounds on a
single operator assuming all others are zero, while the marginalised limits
allow all other coefficients to vary. These can respectively be taken as
optimistic and pessimistic estimates of the sensitivity, and we expect
realistic models to generate some intermediate subset.

In this Section we go a step further in the interpretation of our fit and
explore how specific UV completions of the SM Lagrangian are constrained by
current measurements. In any given model, the global analysis we have
presented is often not directly applicable, as typical models may generate
more than just one of the dimension-6 SMEFT operators, but not all of them.
Moreover, when a specific model contributes to more than a single operator,
these contributions are often related, corresponding to a smaller subset of
independent parameters.

To illustrate these model-dependent effects we have considered several model
interpretations. In Section 7.1 we discuss models in which SMEFT operators are
induced at the tree level, and in Section 7.2 we discuss classes of UV
completions that share similar SMEFT patterns. We then analyze supersymmetric
models with TeV-scale stops in which SMEFT operators are induced at the one-
loop level in Section 7.3. Finally, in Section 7.4 we present results from a
survey of the pulls for all fits with non-vanishing coefficients for
combinations of 2, 3, 4 and 5 operators.

###  7.1 Simple tree-level-induced SMEFTs

We first study the implications of our analysis for single-field extensions of
the SM that contribute to the SMEFT at tree level. This exercise updates the
one presented in Ellis:2018gqa  and is based on the dictionary provided in
deBlas:2017xtg  141414We note that these one-parameter extensions of the SM
have been included among the BSM benchmark proposals made by the LHC Higgs
Working Group Marzocca:2020jze ..

Name | Spin | SU(3) | SU(2) | U(1) | Name | Spin | SU(3) | SU(2) | U(1)
---|---|---|---|---|---|---|---|---|---
$S$ | 0 | 1 | 1 | 0 | $\Delta_{1}$ | $\frac{1}{2}$ | 1 | 2 | $-\frac{1}{2}$
$S_{1}$ | 0 | 1 | 1 | 1 | $\Delta_{3}$ | $\frac{1}{2}$ | 1 | 2 | $-\frac{1}{2}$
$\varphi$ | 0 | 1 | 2 | $\frac{1}{2}$ | $\Sigma$ | $\frac{1}{2}$ | 1 | 3 | 0
$\Xi$ | 0 | 1 | 3 | 0 | $\Sigma_{1}$ | $\frac{1}{2}$ | 1 | 3 | -1
$\Xi_{1}$ | 0 | 1 | 3 | 1 | $U$ | $\frac{1}{2}$ | 3 | 1 | $\frac{2}{3}$
$B$ | 1 | 1 | 1 | 0 | $D$ | $\frac{1}{2}$ | 3 | 1 | $-\frac{1}{3}$
$B_{1}$ | 1 | 1 | 1 | 1 | $Q_{1}$ | $\frac{1}{2}$ | 3 | 2 | $\frac{1}{6}$
$W$ | 1 | 1 | 3 | 0 | $Q_{5}$ | $\frac{1}{2}$ | 3 | 2 | $-\frac{5}{6}$
$W_{1}$ | 1 | 1 | 3 | 1 | $Q_{7}$ | $\frac{1}{2}$ | 3 | 2 | $\frac{7}{6}$
$N$ | $\frac{1}{2}$ | 1 | 1 | 0 | $T_{1}$ | $\frac{1}{2}$ | 3 | 3 | $-\frac{1}{3}$
$E$ | $\frac{1}{2}$ | 1 | 1 | -1 | $T_{2}$ | $\frac{1}{2}$ | 3 | 3 | $\frac{2}{3}$
$T$ | $\frac{1}{2}$ | 3 | 1 | $\frac{2}{3}$ | $TB$ | $\frac{1}{2}$ | 3 | 2 | $\frac{1}{6}$
Table 3: Single-field extensions of the SM constrained by our analysis.

We list in Table 3 the SU(3)$\times$SU(2)$\times$U(1) quantum numbers of the
new fields considered here. We assume flavour-universal couplings in all cases
except $T$ and $TB$, in which the new fields couple only to the third
generation. These two models are taken from Dawson:2020oco . In addition to
evaluating the constraints on these single-field extensions, we also consider
the following two combinations of the fields in Table 3 that yield single-
parameter models via cancellations (see Ref. Marzocca:2020jze ), i.e., models
that depend on only a single coupling ($\lambda$ or $g_{H}$), as well as a
mass $M$: 1) Quark bidoublet model: $\\{Q_{1},Q_{7}\\}$ with equal masses $M$
and equal couplings $\lambda$ to the top quark, and 2) Vector-singlet pair
model: $\\{B,B_{1}\\}$ with equal masses $M$ and Higgs couplings proportional
to $g_{H}$. We exhibit in Tables 4 and 5 the contributions made by exchanges
of each of these fields to the SMEFT coefficients. The numbers shown in the
Tables should each be multiplied by the appropriate squared coupling factors
and divided by the square of the mass scale $M$.

Model | $C_{HD}$ | $C_{ll}$ | $C_{Hl}^{3}$ | $C_{Hl}^{1}$ | $C_{He}$ | $C_{H\Box}$ | $C_{\tau H}$ | $C_{tH}$ | $C_{bH}$
---|---|---|---|---|---|---|---|---|---
$S$ |  |  |  |  |  | $-1$ |  |  |
$S_{1}$ |  | 1 |  |  |  |  |  |  |
$\Sigma$ |  |  | $\frac{5}{8}$ | $\frac{3}{16}$ |  |  | $\frac{y_{\tau}}{4}$ |  |
$\Sigma_{1}$ |  |  | $-\frac{5}{8}$ | $-\frac{3}{16}$ |  |  | $\frac{y_{\tau}}{8}$ |  |
$N$ |  |  | $-\frac{1}{4}$ | $\frac{1}{4}$ |  |  |  |  |
$E$ |  |  | $-\frac{1}{4}$ | $-\frac{1}{4}$ |  |  | $\frac{y_{\tau}}{2}$ |  |
$\Delta_{1}$ |  |  |  |  | $\frac{1}{2}$ |  | $\frac{y_{\tau}}{2}$ |  |
$\Delta_{3}$ |  |  |  |  | $-\frac{1}{2}$ |  | $\frac{y_{\tau}}{2}$ |  |
$B_{1}$ | $1$ |  |  |  |  | $-\frac{1}{2}$ | $-\frac{y_{\tau}}{2}$ | $-\frac{y_{t}}{2}$ | $-\frac{y_{b}}{2}$
$\Xi$ | $-2$ |  |  |  |  | $\frac{1}{2}$ | $y_{\tau}$ | $y_{t}$ | $y_{b}$
$W_{1}$ | $-\frac{1}{4}$ |  |  |  |  | $-\frac{1}{8}$ | $-\frac{y_{\tau}}{8}$ | $-\frac{y_{t}}{8}$ | $-\frac{y_{b}}{8}$
$\varphi$ |  |  |  |  |  |  | $-y_{\tau}$ | $-y_{t}$ | $-y_{b}$
$\\{B,B_{1}\\}$ |  |  |  |  |  | $1$ | $y_{\tau}$ | $y_{t}$ | $y_{b}$
$\\{Q_{1},Q_{7}\\}$ |  |  |  |  |  |  |  | $y_{t}$ |
Model | $C_{HG}$ | $C_{Hq}^{3}$ | $C_{Hq}^{1}$ | $(C_{Hq}^{3})_{33}$ | $(C_{Hq}^{1})_{33}$ | $C_{Hu}$ | $C_{Hd}$ | $C_{tH}$ | $C_{bH}$
$U$ |  | $-\frac{1}{4}$ | $\frac{1}{4}$ | $-\frac{1}{4}$ | $\frac{1}{4}$ |  |  | $\frac{y_{t}}{2}$ |
$D$ |  | $-\frac{1}{4}$ | $-\frac{1}{4}$ | $-\frac{1}{4}$ | $-\frac{1}{4}$ |  |  |  | $\frac{y_{b}}{2}$
$Q_{5}$ |  |  |  |  |  |  | $-\frac{1}{2}$ |  | $\frac{y_{b}}{2}$
$Q_{7}$ |  |  |  |  |  | $\frac{1}{2}$ |  | $\frac{y_{t}}{2}$ |
$T_{1}$ |  | $-\frac{5}{8}$ | $-\frac{3}{16}$ | $-\frac{5}{8}$ | $-\frac{3}{16}$ |  |  | $\frac{y_{t}}{4}$ | $\frac{y_{b}}{8}$
$T_{2}$ |  | $-\frac{5}{8}$ | $\frac{3}{16}$ | $-\frac{5}{8}$ | $\frac{3}{16}$ |  |  | $\frac{y_{t}}{8}$ | $\frac{y_{b}}{4}$
$T$ | $-\frac{M_{T}^{2}}{v^{2}}\frac{\alpha_{s}(0.02)}{8\pi}$ |  |  | $-\frac{1}{2}\frac{M_{T}^{2}}{v^{2}}$ | $\frac{1}{2}\frac{M_{T}^{2}}{v^{2}}$ |  |  | $y_{t}\frac{M_{T}^{2}}{v^{2}}$ |
Table 4: Operators generated by the tree-level single-field extensions listed in the first column. Each extension depends on a single coupling $\lambda$ as well as a new physics mass-scale $M$. The coefficients of the operators are each proportional to the squares of the corresponding coupling $\lambda$ by the corresponding entry in the Table and divided by $M^{2}$. $y_{t}$, $y_{b}$ and $y_{\tau}$ denote the top, bottom and tau Yukawa couplings respectively, $v$ denotes the electroweak scale and $\alpha_{s}$ denotes the strong coupling. Model | $C_{HD}$ | $C_{H\Box}$ | $C_{\tau H}$ | $C_{tH}$ | $C_{bH}$
---|---|---|---|---|---
$B$ | $-2a^{2}$ | $-\frac{1}{2}(a^{2}-b^{2})$ | $-aby_{\tau}$ | $-aby_{t}$ | $-aby_{b}$
$W$ | $\frac{1}{2}b^{2}$ | $-\frac{1}{8}(3a^{2}+b^{2})$ | $-\frac{1}{4}y_{\tau}(a+b)^{2}$ | $-\frac{1}{4}y_{t}(a+b)^{2}$ | $-\frac{1}{4}y_{b}(a+b)^{2}$
Model | $C_{tH}$ | $C_{bH}$ | $C_{Ht}$ | $C_{HG}$ |
$TB$ | $\frac{M_{TB}^{2}}{v^{2}}y_{t}a^{2}$ | $\frac{M_{TB}^{2}}{v^{2}}y_{b}b^{2}$ | $-\frac{M_{TB}^{2}}{v^{2}}a^{2}$ | $-\frac{M_{TB}^{2}}{v^{2}}\frac{\alpha_{s}(0.65)}{8\pi}b^{2}$ |
Table 5: Operator coefficients generated by the tree-level single-field models
$B$, $W$ and $TB$, which each depend on two couplings $a$ and $b$, with
$y_{t}$, $y_{b}$ and $y_{\tau}$ denoting the top, bottom and tau Yukawa
couplings respectively, $v$ denoting the electroweak scale and $\alpha_{s}$
denoting the strong coupling. The coefficients of all operators are
proportional to the corresponding entries in the Table and divided by $M^{2}$.

We show in Fig. 14 the results from our global fit for all of the one-
parameter single-field extensions of the SM. In each of these models we
constrain a positive quantity: $|\lambda|^{2}$. The constraints in Fig. 14 are
found using the numerical MCMC fitter described in Appendix B. This method
allows us to incorporate the constraint $|\lambda|^{2}>0$ as a Heaviside prior
$\pi(|\lambda|^{2}<0)=0$. The 2-$\sigma$ constraints on the mass scales in TeV
units, assuming that the corresponding couplings are set to unity, are shown
as horizontal bars 151515In the case of the $T$ (vector-like quark) model, the
mass limit has been obtained using the relation $s^{t}_{L}\simeq\lambda
v/\sqrt{2}M_{T}$ and setting $\lambda=1$ Dawson:2020oco .. We note that most
of these limits exceed 1 TeV for unit coupling and do not depend on kinematic
distributions probing this region, in which case the SMEFT approach is self-
consistent. The SU(2)-singlet VLQ top-partner model ($T$) and $S$ are the most
poorly constrained, with $m_{S,T}>$900, 770 GeV. We also show in grey boxes
the corresponding bounds on the squared couplings, assuming a mass scale of 1
TeV. Most of the bounds are $<1$, justifying a tree-level treatment. We also
list all the pulls that exceed 1-$\sigma$, which is significant.

Figure 14: The horizontal bars show the mass limits (in TeV) at the 95% CL for
the models described in Table 3, setting the corresponding couplings to unity.
The coupling limits obtained when setting the mass to 1 TeV are shown in grey
boxes. We also note in light blue the pulls that exceed 1-$\sigma$.

We can compare the mass limits for these models with the naive scale limits
shown in Fig. 6. In a model-independent SMEFT analysis, one allows all the EFT
operators to vary simultaneously. On the other hand, in specific models not
all EFT coefficients are generated and those that do appear are related to
each other in such a way that the number of free parameters of the model is
matched to the number of independent EFT operators generated by the model. As
an example, we discuss the following set of single-parameter models:

$\Sigma$, $\Sigma_{1}$, $N$ and $E$ ,

which all span the same types of SMEFT operators. They are all characterized
by non-zero values for the following set of operators involving electroweak
precision lepton observables:

$C_{H\ell}^{1,3}\neq 0$ ,

whereas the other operator coefficients are zero, or very mildly constrained
(e.g., $C_{\tau H}$, which is $\propto y_{\tau}$). Interpreted in terms of
these models, the global SMEFT fit leads to mass limits of the order of 5 TeV
for unit couplings, or corresponding coupling limits of ${\cal O}(10^{-1})$
for TeV resonances. Each of these particles generates a different relation
between $C_{H\ell}^{1}$ and $C_{H\ell}^{3}$, leading to slightly different
limits, e.g., for the model with a new neutral fermion $N$ one expects
$C_{H\ell}^{1}=-C_{H\ell}^{3}=|\lambda_{N}|^{2}/4m_{N}^{2}$. The sizes of the
mass limits justify the SMEFT approach, and the small couplings for masses of
a TeV justify working at tree level.

The constraints on particles beyond the SM would be weaker if their effects
were not tree-level but loop-induced. These are typical of extensions of the
SM where couplings to new states have to be in pairs, as would be the case if
they carry a new conserved quantum number. We discuss in Section 7.3 one
particularly interesting example with such loop-induced effects, namely stops
in an $R$-parity-conserving supersymmetric model.

###  7.2 Tree-level SMEFT patterns

As already commented, and displayed in Table 4, simple extensions of the SM
exhibit specific patterns in the operators they generate. Many operators have
vanishing coefficients and those that are non-zero are often related. For
example, model $B_{1}$ of Table 3 generates the bosonic operators $C_{H\Box}$,
$C_{HD}$ and Yukawa operators such as $C_{tH}$. These three operators are
related by $C_{H\Box}=C_{tH}=-\frac{1}{2}C_{HD}$. Similarly, model $W_{1}$
generates a pattern $C_{H\Box}=C_{tH}=\frac{1}{2}C_{HD}$. Motivated by these