DESY 17-231

Diboson at the LHC vs LEP

Christophe<EMAIL_ADDRESS>Marc
<EMAIL_ADDRESS>Marc<EMAIL_ADDRESS>
a DESY, Notkestrasse 85, 22607 Hamburg, Germany
b Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
c IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona, Spain
d Départment de Physique Théorique, Université de Genève, Genève, Switzerland

We use the current CMS and ATLAS data for the leptonic $pp\to WW,WZ$ channels
to show that diboson production is, for a broad class of flavour models,
already competitive with LEP-1 measurements for setting bounds on the
dimension six operators parametrising the anomalous couplings between the
quarks and the electroweak gauge bosons, at least under the assumption that
any new particle is heavier than a few TeV. We also make an estimate of the
HL-LHC reach with $3$ ab-1. We comment on possible BSM interpretations of the
bounds, and show the interplay with other searches for a simplified model with
vector triplets. We further study the effect of modified $Z$-quark-quark
couplings on the anomalous triple gauge coupling bounds. We find that their
impact is already significant and that it could modify the constraints on
$\delta g_{1z}$ and $\delta\kappa_{\gamma}$ by as much as a factor two at the
end of HL-LHC ($\lambda_{z}$ is only marginally affected), requiring a global
fit to extract robust bounds. We stress the role of flavour assumptions and
study explicitly flavour universal and minimal flavour violation scenarios,
illustrating the differences with results obtained for universal theories.

###### Contents

  1. 1 Introduction
  2. 2 Theoretical framework
    1. 2.1 High energy behaviour and correlations
    2. 2.2 Helicity amplitudes at high energy and correlations between aTGC and $\delta V\bar{q}q$
  3. 3 Results with current LHC data
    1. 3.1 Data used and statistical analysis
    2. 3.2 Correlations among the Higgs basis parameters
    3. 3.3  $\delta V\bar{q}q$: LHC bounds vs LEP-1 constraints
    4. 3.4 LHC bounds on aTGC and interplay with $\delta V\bar{q}q$
  4. 4 Projected bounds for HL-LHC
    1. 4.1 Data used and assumptions for HL-LHC
    2. 4.2 HL-LHC projections on $\delta V\bar{q}q$ vs LEP-1
    3. 4.3 HL-LHC projections on aTGC and interplay with $\delta V\bar{q}q$
  5. 5 Interpretation of the constraints
    1. 5.1 Quadratic BSM amplitudes
    2. 5.2 Power countings and BSM interpretations
    3. 5.3 Energy limitation
    4. 5.4 A model with triplets: diboson reach vs other searches
  6. 6 Summary and outlook
  7. A Correlations in the Higgs basis
  8. B Summary of LEP-1 bounds
  9. C Cross checks of the aTGC bounds
  10. D Comparison of HEP parameter bounds
  11. E High energy amplitudes in the Warsaw basis
  12. F Summary tables

##  1 Introduction

The Large Hadron Collider (LHC) is probing the Standard Model (SM) at higher
energies than ever before, reaching new regions never explored so far. For
this reason, we must take the chance to learn as much as possible from it.
With the discovery of a scalar particle consistent with the Higgs boson, the
SM can in principle be consistent up to the Planck scale. Nonetheless in many
UV completions predicting a light Higgs, e.g. supersymmetric or composite
Higgs models, one requires other new particles with masses around the
electroweak scale unless there is some fine tuning. So far though, the LHC has
not seen any robust hints of new physics, which indicates that any new
particles must be either too weakly coupled to the SM or heavy enough to not
have been seen. In the first case, one may expect to see direct effects, like
for example a resonance showing up once enough luminosity is collected. In the
second case, one expects to see the effects of the new particles indirectly,
for example, by modifying the differential cross sections of particular
processes with respect to the SM prediction.

The study of diboson production, $pp\to WV,V=W,Z$, offers a way to probe
physics scenarios of the second class. The interest in such channels both at
lepton and hadron colliders is not new [1, 2, 3] but it has recently received
renewed attention, see in particular Refs. [4, 5, 6, 8, 11, 9, 7, 10, 12, 13,
14]. This is first due to the fact that together with $pp\to Vh$ [15, 16],
diboson production directly probes the interactions of the Goldstone bosons
via the gauge boson longitudinal polarizations, and therefore is one of the
first places where to expect signs of new physics related to the electroweak
symmetry breaking. Furthermore, in Refs. [6, 12, 10] it has been shown that at
high energy, the leading amplitudes for $pp\to VV,Vh$ grow with the center of
mass energy faster than the SM ones and therefore diboson production can
benefit from the higher energy probed at the LHC to reveal sign of new
physics. See Refs. [17, 18, 9, 19, 15, 20, 21, 12, 22, 23] for studies using
this high energy behaviour to increase the sensitivity to $d=6$ operators;
notice that in some cases the new LHC bounds can improve on the LEP-1 and
LEP-2 bounds. As shown in Refs. [6, 12, 10, 24], in the Higgs basis and at the
dimension-six level [25, 26], there are a priori seven Beyond the Standard
Model (BSM) coefficients that modify the diboson amplitude $pp\to WV$ at high
energies. These are three anomalous triple gauge couplings (aTGC),
traditionally parametrised by $\delta\kappa_{\gamma},\,\delta
g_{1z},\,\lambda_{\gamma}$, and four anomalous couplings between the light
quarks and the $Z$ gauge boson, $\delta g_{L}^{Zu}$, $\delta g_{R}^{Zu}$,
$\delta g_{L}^{Zd}$, $\delta g_{R}^{Zd}$ ($\delta V\bar{q}q$ hereinafter),
that will be introduced later. Interestingly, the $pp\to VV$ and $pp\to Vh$
amplitudes at high energy become equal, as expected by the Goldstone
equivalence theorem, and actually only depend on five combinations of the
$d=6$ operators [6, 12, 10]. There has not been yet a complete global analysis
establishing the future bounds on these five independent so-called High Energy
Parameters, but some first results have been obtained in the $WZ$ [12] and
$Zh$ [16] channels, showing some nice complementarity. Combining with LEP
constraints on the Z couplings to fermions, one could in principle univocally
derive bounds on the aTGCs. The purpose of our work is to stress that, if this
strategy is perfectly fine for universal theories, the aTGC bounds obtained
that way do not directly apply when other flavour assumptions are considered
and one needs to perform a global fit to derive bounds on both aTGCs and
$V\bar{q}q$ couplings.

In this work, we study the constraining power of diboson data to set bounds on
the anomalous couplings between the $W$ and $Z$ gauge bosons and the light
quarks. In particular, we make use of the differential distributions reported
by the experimental collaborations. We find that due to the enhanced
sensitivity at high energies, $pp\to WV$ can already be competitive or even
surpass LEP-1 on setting bounds on $\delta V\bar{q}q$, at least under the
assumptions that these anomalous couplings are generated by new particles with
masses equal or greater than a few TeV to ensure the validity of the EFT, see
section 5.1. We refer the reader to Refs. [4, 12, 9, 11, 23] where new
differential distributions and experimental searches are proposed in order to
increase the sensitivity to the effective field theory (EFT) operators
entering diboson production. If these are implemented by the experiments, the
increase of sensitivity could allow diboson production at HL-LHC to set much
stringent bounds on the BSM amplitudes, reaching the point where they are
smaller than the SM, and therefore can start constraining BSM scenarios with a
characteristic coupling smaller than a typical SM gauge coupling.

We rely on the differential distributions from the up-to-date diboson
measurements performed by ATLAS and CMS with up to 20 fb-1 of data at 8 TeV
and 13 fb-1 of data at 13 TeV, see Table 1 for details. We also estimate the
sensitivity expected at the high-luminosity run of the LHC (HL-LHC) with an
anticipated total of 3 ab-1 of data. We consider two general flavour
structures of the higher dimensional operators: i) Flavour Universality (FU),
where the EFT operators satisfy a $U(3)^{5}$ family symmetry, which, in the
Higgs basis, corresponds to $[\delta g_{L,R}^{Zu,d}]_{ij}=A_{L,R}^{u,d}\
\delta_{ij}$, and ii) Minimal Flavour Violation (MFV) where this symmetry is
broken only by spurions of the Yukawa couplings, leaving $[\delta
g_{L,R}^{Zu,d}]_{ij}\simeq\left(A_{L,R}^{u,d}+B_{L,R}^{u,d}\
\frac{m_{i}^{2}}{m_{3}^{2}}\right)\delta_{ij}\,$. 111When simulating the
diboson production, we only modify the couplings to the $u$ and $d$ quark
since the BSM effects from the heavier quarks are PDF-negligible, below 1%,
within the flavour assumptions considered. Therefore the diboson analysis
presented in this paper does not distinguish MVF and $U(2)^{5}$-flavour
symmetric setups. It would also be interesting to combine and compare the
LEP-1 bounds on the $\delta V\bar{q}q$ couplings with other flavour scenarios,
e.g. the anarchic case (see Ref. [27]) or the diagonal one, i.e. diag$({\delta
g_{L,R}^{Zu,d}}_{11},{\delta g_{L,R}^{Zu,d}}_{22},\,{\delta
g_{L,R}^{Zu,d}}_{33})$. We leave these analysis for future work since the non-
Gaussianity of the fit makes it non-trivial to go from a more general case to
a more restrictive one. Diboson production at hadron colliders is insensitive
to these assumptions since the cross section is dominated by the light quarks,
while the constrains from LEP-1 can change by an order of magnitude, see the
results of Ref. [27] that we summarise in Appendix B. Another interesting UV
assumption is that of universal theories [28, 29, 30]. These can be defined as
those theories whose EFT can be fully described by bosonic operators and
deviations of the light quark couplings can be written in terms of the gauge
boson oblique parameters. Given that the LEP-1 bounds for these types of
theories is one or two orders of magnitude stronger than those for MFV and FU,
we found that with the current experimental searches diboson production is not
competitive with LEP-1 for unviersal theories. 222See Appendix B for the
bounds on universal theories expressed in the Higgs basis.

Due to the larger systematics at the LHC, the conclusion that it can surpass
LEP-1 and LEP-2 in setting bounds on the EFT operators may come as a surprise,
but it follows from the fact that some BSM amplitudes can grow with the
characteristic scale of the hard process probed at the LHC. However, it is
important to keep in mind, as stressed in Refs. [17, 15, 20, 21, 12], that the
larger systematics also imply in many cases that the new LHC bounds are valid
only when the BSM contribution is larger than the SM one, limiting in some
cases the generality of these bounds to subsets of possible UV theories. We
comment more on the EFT interpretation in section 5. Nonetheless, given that
the LHC is running and we do not know what new physics may lie ahead, it is
still important to make sure that all the regions of the EFT parameter space
are explored in the most model independent way as possible.

Besides studying the bounds on $\delta V\bar{q}q$, we also look at the impact
of non-vanishing $\delta V\bar{q}q$ in the aTGC determination under the
different flavour assumptions considered. Looking at this effect was first
mentioned and motivated in Refs. [31, 7] and checked explicitly in Ref. [10]
using the channel $pp\to WW$ at 8 TeV by ATLAS. We extend this analysis by
first performing a global fit to the present data for all the channels in
Table 1 and also by studying the impact of different flavour assumptions. We
also estimate the sensitivity that one can hope to reach at HL-LHC, concluding
that the effect of $\delta V\bar{q}q$ will be more and more important in the
future. It should be noted that the analysis we provide is done at leading
order (LO). We expect that the NLO effects are most relevant for amplitudes
with final transverse polarizations due to the non-interference effects shown
in Ref. [32, 10]. The NLO effects can also be relevant in certain regions of
the phase space; for instance in the amplitude of $pp\to WZ$ which nearly
vanishes for the $\pm\mp$ and $\pm 0$ polarizations when the polar scattering
angle is $\theta\simeq\pi/2$ [12]. Since in our study we are mostly interested
in the cases where the two gauge bosons have longitudinal polarizations, we do
not expect much difference in our conclusions even though it would be
interesting to study in more detail the NLO effects.

We briefly comment on possible interpretations of the EFT bounds derived. To
gain perspective and a sense of the usefulness of the constraints coming from
diboson production, we study a simplified model of heavy vector triplets and
compare the diboson bounds to the ones from other searches like dijets,
resonant diboson or Higgs coupling measurements, finding that diboson can be
complementary to other processes in exploring the parameter space of the
model, and can be the leading probe in important regions of parameters.

The paper is organized as follows. In section 2, we give the conventions and
review the high energy behavior of diboson production at the LHC. In
particular, we note that the high-energy diboson amplitudes in the Higgs basis
are controlled by seven independent parameters in FU and MFV setups as opposed
to five parameters only for universal theories. In section 3, we present the
bounds on the $\delta V\bar{q}q$ and the effect of allowing these to be non-
zero in the aTGC exclusion plots. In section 4, we estimate the $\delta
V\bar{q}q$ bounds that can be expected by the end of HL-LHC and we quantify
the effect of letting $\delta V\bar{q}q$ and aTGC float in global EFT fit. In
section 5, we briefly review the validity of the EFT approach in presence of
non-negligible contributions from the dimension six BSM quadratic amplitudes,
and we review various UV scenarios and power counting rules which motivate the
various assumptions on the values of the parameters used through the paper. We
also study a toy model with heavy triplets as a concrete example. And we
compare our HL-LHC bounds for this toy model with those coming from Higgs
coupling measurements and dijet searches. We conclude in section 6. Four
appendices provide further technical details and cross-checks.

##  2 Theoretical framework

We work in the so called Higgs basis [24, 25, 26], and follow the conventions
of Ref. [25] where $\alpha,G_{F}$ and $m_{Z}$ are taken as the input
parameters. The Higgs basis parametrizes the $d=6$ EFT operators as
modifications to the SM vertices, and where the fields are in the mass
eigenstates and in the unitary gauge. In this basis and considering only
operators with $d\leq 6$, the relevant terms for $pp\to WV$ production are:

${\cal L}_{\rm diboson}\supset{\cal L}_{\rm TGC}+{\cal L}_{V\bar{q}q}\,.$ (1)

The first term contains the SM interactions between the electroweak gauge
bosons together with the $d=6$ aTGC deformations,

$\displaystyle{\cal L}_{\rm TGC}$ $\displaystyle=$ $\displaystyle
ie\left(W_{\mu\nu}^{+}W_{\mu}^{-}-W_{\mu\nu}^{-}W_{\mu}^{+}\right)A_{\nu}+ie\left[(1+\delta\kappa_{\gamma})A_{\mu\nu}\,W_{\mu}^{+}W_{\nu}^{-}\right]$
(2) $\displaystyle+$ $\displaystyle ig\,c_{W}\left[(1+\delta
g_{1,z})\left(W_{\mu\nu}^{+}W_{\mu}^{-}-W_{\mu\nu}^{-}W_{\mu}^{+}\right)Z_{\nu}+(1+\delta\kappa_{z})\,Z_{\mu\nu}\,W_{\mu}^{+}W_{\nu}^{-}\right]$
$\displaystyle+$ $\displaystyle i{e\over
m_{W}^{2}}\lambda_{\gamma}W_{\mu\nu}^{+}W_{\nu\rho}^{-}A_{\rho\mu}+i{g\,c_{W}\over
m_{W}^{2}}\lambda_{z}W_{\mu\nu}^{+}W_{\nu\rho}^{-}Z_{\rho\mu}\,.$

The second term in the Lagrangian (1) contains the SM contribution and
deviations to the couplings between the up and down quarks to the $W,\,Z$,
gauge bosons,

$\displaystyle{\cal L}_{V\bar{q}q}$ $\displaystyle=$
$\displaystyle\sqrt{g^{2}+g^{\prime}{}^{2}}Z_{\mu}\left[\sum_{f\in
u,d}\bar{f}_{L}\gamma_{\mu}\left(T^{3}_{f}-s_{W}^{2}Q_{f}+\delta
g^{Zf}_{L}\right)f_{L}+\sum_{f\in
u,d}\bar{f}_{R}\gamma_{\mu}\left(-s_{W}^{2}Q_{f}+\delta
g^{Zf}_{R}\right)f_{R}\right]$ (3) $\displaystyle+$
$\displaystyle{g\over\sqrt{2}}\left(W_{\mu}^{+}\bar{u}_{L}\gamma_{\mu}\left(I_{3}+\delta
g^{Wq}_{L}\right)d_{L}+{\mathrm{h}.c.}\right)\,.$

Since at dimension six the following relations are satisfied (see for instance
Ref. [25]):

$\delta\kappa_{z}=\delta
g_{1}^{z}-\tan^{2}\theta\,\delta\kappa_{\gamma}\,,\hskip
36.98866pt\lambda_{z}=\lambda_{\gamma}\,,\hskip 36.98866pt\delta
g^{Wq}_{L}=\delta g^{Zu}_{L}-\delta g^{Zd}_{L}\,,$ (4)

the deviations of the schematic form $\sim g_{SM}\,(1+\delta$) can be
parametrized by two independent aTGC (which we choose to be
$\delta\kappa_{\gamma}$, $\delta g_{1z}$), and four independent corrections to
$Z\bar{q}q$ vertices (which we choose to be $\delta g_{L}^{Zu}$, $\delta
g_{R}^{Zu}$, $\delta g_{L}^{Zd}$, $\delta g_{R}^{Zd}$). Notice that the aTGC
parametrized by $\lambda_{\gamma}=\lambda_{z}$ introduces a new type of
coupling non-existent in the SM. In total there are, in the Higgs basis, seven
parameters that contribute to the leading deformations to diboson production
(three aTGC and four $\delta V\bar{q}q$).

In the Lagrangian (3) we have not included right-handed charged currents nor
dipole contributions since under FU and MFV they are either zero, or are
suppressed by the Yukawas of the light quarks. We also ignore the deformations
in the lepton sector since their bounds from LEP-1 data are an order of
magnitude better than those on the quark sector [27]. Finally we also ignored
the shift to the $W$ mass, $\delta_{m}$, since its current existing bound is
such that it numerically gives in diboson production an effect ten times
smaller than a modified quark coupling.

###  2.1 High energy behaviour and correlations

In the Higgs basis, the energy growth of the amplitudes that interfere with
the SM in the high energy limit can be understood as follows. At tree level in
the SM, and in the unitary gauge, the leading amplitude for
$q\bar{q}^{\prime}\to WW\,(WZ)$ is given by the sum of three diagrams,
consisting of an s-channel exchange of the $\gamma,\,Z$ bosons ($W$ boson),
and a t-channel contribution. Taking as an example the case of $\bar{q}q\to
WW$, where $\bar{q}q=\bar{u}u,\,\bar{d}d$, one finds that the tree level SM
amplitude is given by the Feynman diagrams of Fig. -899. One can check that at
large center of mass energy, $\hat{s}\gg m_{W}^{2}$, the total amplitude for
$q\bar{q}\to W^{+}_{0}W^{-}_{0}$ is given by [33]

$\mathcal{M}_{\gamma}+\mathcal{M}_{Z}+\mathcal{M}_{t}=i\,\hat{s}\left[-\frac{e^{2}\,\sin\theta}{2m_{W}^{2}}Q_{q}\,-\frac{e^{2}\,\sin\theta}{2m_{W}^{2}}\frac{1}{s_{W}^{2}}(T_{q}^{3}-s_{W}^{2}Q_{q})\
+\frac{e^{2}\,\sin\theta}{2m_{W}^{2}}\frac{T_{q}^{3}}{s_{W}^{2}}\right]+\cdots$
(5)

where $W^{\pm}_{0}$ stand for the longitudinal polarizations of the $W^{\pm}$
gauge bosons, $\hat{s}$ is the squared center of mass energy, the dots denote
sub-leading contributions at high energy, $Q_{q}$ and $T^{3}_{q}$ are the
electric charge and $SU(2)$ weak isospin of the initial quarks and $\theta$ is
the angle between $W^{+}$ and the beam axis. 333There is another term that
grows with energy but we neglected it since it is proportional to the quark
masses. Its energy growth is canceled with the diagram including the Higgs,
however, in our energy range it is negligible, and as a first approximation
one can think of the quarks to be massless.

The key point of Eq. (5) is to notice that while each of the individual sub-
amplitudes grows with $\hat{s}$, the sum does not. Therefore, any shift to the
SM couplings, shown in blue and red in Fig. -899, will spoil the cancellation
of the different pieces in Eq. (5), and therefore the resulting amplitude will
be proportional to $\hat{s}$. In the Higgs basis it is especially clear to see
that all the coefficients modifying diboson production with a shift to the SM
couplings will generically induce an amplitude that grows with $\hat{s}$.

Figure -899: Representative contributions to diboson production. The
sensitivity of the measurements, already with
$\mathcal{O}(20)\,\textrm{fb}{}^{-1}$ of data and certainly even more at HL-
LHC, is such that they can improve the LEP-1 constraints on the quark
couplings to gauge bosons (blue). This also implies that the LEP-1 bounds are
no longer stringent enough to make these parameters negligible when setting
bounds on the anomalous triple gauge couplings (red).

Notice that the interaction given by
$\lambda_{\gamma}W_{\mu\nu}^{+}W_{\nu\rho}^{-}(s_{W}A_{\rho\mu}+c_{W}Z_{\rho\mu})$
in Eq. (3) is not present in the SM. In this case one cannot use the spoiling
of the SM amplitude cancellation of Eq. (5) to see whether its effect
asymptotically grows with the center of mass energy. Nonetheless, one can see
by direct calculation that the amplitude induced by this operator actually
grows with $\hat{s}$ as a consequence of the presence of extra derivatives in
the interaction.

###  2.2 Helicity amplitudes at high energy and correlations between aTGC and
$\delta V\bar{q}q$

To estimate which operators or combinations of operators will be the most
constrained by diboson production at the LHC, one can study each helicity
amplitude as done in Refs. [6, 12]. In the limit where $\hat{s}\gg m_{W}^{2}$,
the leading helicity amplitudes for the partonic scattering $q\bar{q}\to WW$
are given by 444The $L,R$ stand for the initial helicities of the quarks,
while $\pm$ and $0$ stand for the transverse and longitudinal polarizations of
the final electroweak bosons respectively. We computed these amplitudes using
FeynCalc [34] using the BSMC package [35] for FeynRules [36], finding
agreement with the expressions presented in Ref. [6], which also was a cross
check for the .ufo file used in the Madgraph5 simulations.

$\displaystyle\mathcal{M}(LL;00)$ $\displaystyle=$ $\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}\,\sin\theta}{2s_{W}^{2}}\left[(2T_{q}^{3})\,\delta
g_{L}^{Wq}-\delta g_{L}^{Zq}-\delta
g_{1z}(T_{q}^{3}-s_{W}^{2}Q_{q})+\delta\kappa_{\gamma}t_{W}^{2}(T_{q}^{3}-Q_{q})\right]$
$\displaystyle\mathcal{M}(RR;00)$ $\displaystyle=$ $\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}\,\sin\theta}{2s_{W}^{2}}\left[\delta
g_{R}^{Zq}-\delta
g_{1z}\,s_{W}^{2}\,Q_{q}+\delta\kappa_{\gamma}\,t_{W}^{2}\,Q_{q}\right]$
$\displaystyle\mathcal{M}(LL;\pm\pm)$ $\displaystyle=$ $\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}\,\sin\theta}{2s_{W}^{2}}\,T_{q}^{3}\lambda_{\gamma}\,,$
(6)

where $\delta g^{Wq}_{L}=\delta g^{Zu}_{L}-\delta g^{Zd}_{L}$ and $\delta
g_{L,R}^{Zq}$ corresponds to the anomalous vertex of the incoming quark $q$,
defined in Eq. (3). For $qq^{\prime}\to WZ$, the energy growing amplitudes are

$\displaystyle\mathcal{M}(LL;00)$ $\displaystyle=$
$\displaystyle-i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}\,\sin\theta}{2\sqrt{2}s_{W}^{2}c_{W}}\left[\delta
g_{L}^{Zu}-\delta g_{L}^{Zd}-\delta g_{1z}c_{W}^{2}\right]$
$\displaystyle\mathcal{M}(LL;\pm\pm)$ $\displaystyle=$
$\displaystyle-i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}\,\sin\theta}{2\sqrt{2}s_{W}^{2}c_{W}}\,\lambda_{z}.$
(7)

We can see, as pointed out in Refs. [6, 12], that in the asymptotic high
energy regime there are only five independent combinations of parameters
entering $pp\to WV$ since

$\mathcal{M}(u_{L}\bar{d}_{L}\to
W_{0}Z_{0})\,={1\over\sqrt{2}\,c_{W}}\,\left(\mathcal{M}(\bar{d}_{L}d_{L}\to
W_{0}W_{0})-\mathcal{M}(\bar{u}_{L}u_{L}\to W_{0}W_{0})\right)+{\cal
O}(\hat{s}^{0})\,.$ (8)

Therefore, there are only four relevant independent combinations for the
longitudinal polarizations and one for the transverse ones that can be probed
in the high energy limit. 555This counting may change for other flavour
assumptions, since the right-handed charged current, that could be present
away from the FU/MFV setups, gives rise also to an energy-growing amplitude
for $pp\to WZ$:
$\mathcal{M}(RR;00)=-i\frac{e^{2}\,\hat{s}\,\sin\theta}{2\sqrt{2}m_{W}^{2}s_{W}^{2}c_{W}}\,\delta
g_{R}^{Wq}\,+\,\mathcal{O}(\hat{s}^{0})$. These four directions for the
longitudinal polarizations are the so-called High Energy Parameters (HEPs)
introduced in Ref. [12], see Table 2 in this reference. For completeness, in
Appendix D, Eq. (55), these four HEPs are written explicitly in terms of the
Higgs basis [25] parameters. In Appendix E we express the amplitudes shown in
Eqs. (6), (7) in the Warsaw basis.

Notice that if the experimental sensitivity is low such that the quadratic BSM
squared amplitudes dominate the cross section, all the channels above show a
similar behaviour at high-energy. On the other hand, when the experimental
sensitivity is getting good enough to probe BSM deformations subdominant to
the SM, the channels that feature an interference between SM and BSM will be
of better use to bound anomalous couplings. As shown in Refs. [32, 12], this
selects the production of two longitudinally gauge bosons as the preferred
channel. It is nonetheless possible to also use the production of
transversally polarized gauge bosons when relying on specific kinematic
observables to resurrect the interference [9, 11]. In our analysis, which uses
the current experimental observables, we find that the high energy bins are
the most important in setting constrains. For these, we observe that the
quadratic pieces are in general equal or larger than the interference parts,
and therefore the question of the BSM/SM interference is not so relevant in
our analysis.

From Eqs. (6) and (7), there are a total of seven coefficients parameterizing
the five directions growing as $\hat{s}$ in the processes $pp\to WV$. Hence,
in the asymptotic high energy limit, two completely flat directions are
anticipated among the Higgs basis coefficients. A simple way to see the flat
directions explicitly is by noting that any deviation of $\delta g_{1z}$ and
$\delta\kappa_{\gamma}$ in Eqs. (6), (7) can be compensated by a modification
of the vertex corrections $\delta V\bar{q}q$. Naively, if one assumes that the
largest sensitivity comes from the high energy bins, the characteristic energy
of diboson production is $\sqrt{\hat{s}}\sim$ TeV. At these energies, one
expects that the subleading amplitudes, which grow with $\sqrt{\hat{s}}/m_{W}$
instead of $\hat{s}/m_{W}^{2}$, can set bounds that are worse by a factor
$\sqrt{\hat{s}}/m_{W}\sim 10$ (as long as the BSM squared amplitudes dominate
the cross section, which, as we will see, is the case in our analysis). These
subleading amplitudes involve a longitudinal and a transverse vector boson in
the final states. For $pp\to WW$, they are given by

$\displaystyle\mathcal{M}(LL;0\pm)$ $\displaystyle=$
$\displaystyle-\frac{e^{2}\sqrt{\hat{s}}\,\cos^{2}\frac{\theta}{2}}{\sqrt{2}m_{W}s_{W}^{2}}\bigg{[}(2T^{3}_{q})\delta
g_{L}^{Wq}-\delta g_{L}^{Zq}-\frac{1}{c_{W}^{2}}\delta
g_{1z}(T^{3}_{q}-s_{W}^{2}Q_{q})-T_{q}^{3}(\delta\kappa_{\gamma}+\lambda_{\gamma})\bigg{]}\,,$
$\displaystyle\mathcal{M}(RR;0\pm)$ $\displaystyle=$
$\displaystyle-\frac{e^{2}\sqrt{\hat{s}}\,\sin^{2}\frac{\theta}{2}}{\sqrt{2}m_{W}s_{W}^{2}}\left[t_{W}^{2}Q_{q}\delta
g_{1z}-2\delta g_{R}^{Zq}\right],$ (9)

while for $pp\to WZ$, one has

$\displaystyle\mathcal{M}(LL;\pm 0)$ $\displaystyle=$
$\displaystyle-\frac{e^{2}\sqrt{\hat{s}}}{2m_{W}s_{W}^{2}c_{W}}\bigg{[}\delta
g_{L}^{Zu}+\delta g_{L}^{Zd}+(\delta g_{L}^{Zu}-\delta
g_{L}^{Zd})\cos\theta\,-\,c_{W}^{2}(2\delta
g_{1z}+\lambda_{z})\sin^{2}\frac{\theta}{2}\bigg{]},$

up to subleading terms suppressed by $\sim 1/\sqrt{\hat{s}}$. One can check
that the combination of coefficients entering in the subleading amplitudes
cannot be obtained as a linear combination of the directions appearing in the
leading $\hat{s}/m_{W}^{2}$ amplitudes. Hence, one naively expects to find
some directions in the EFT space that are ${\cal{O}}(10)$ times less
constrained than the five directions given by the amplitudes leading at high
energy. We confirm this naive estimate later in section 3.2 where we study the
correlations among the different constraints in the Higgs basis.

To conclude this section, it should be noted that the previous counting is
different for universal theories. As discussed in Appendix B, the high-energy
diboson amplitudes depend only on five independent parameters and no flat
direction is expected in the global fit to diboson data. Anticipating the
results that will be presented in the rest of the paper, one should be aware
nonetheless that LHC diboson data will not be competitive to LEP-1 to
constrain the $Z\bar{q}q$ couplings in universal theories, at least by using
only the current leptonic experimental distributions.

##  3 Results with current LHC data

###  3.1 Data used and statistical analysis

To get the bounds on the different BSM parameters of Eqs. (2)–(3), we have
used all the leptonic channels of the $pp\to WW,\,WZ$ channels reported by CMS
and ATLAS, see Table 1. We indicate in each case the differential distribution
used to perform the combined fit. We limited the analysis to purely leptonic
decays due to their high sensitivity and the ease with which one can reproduce
the experimental analyses. See Ref. [43] for a summary of the ATLAS and CMS
constraints. There are nonetheless other channels that would be interesting to
add, e.g. two quarks and two leptons in the final state [44], since they can
set even tighter constrains than the purely leptonic channels. 666See Ref.
[23] where projections for the semi-leptonic channels at HL-LHC are studied in
detail, and new experimental observables are proposed.

Experiment |  $\mathcal{L}$[fb-1] | $\sqrt{s}$ | Process | Obs. | Ref.
---|---|---|---|---|---
ATLAS | 4.6 | 7 TeV | $WW\to\ell\nu\ell\nu$ | $p_{T\ell}^{(1)}$ |  [37], Fig. 7
ATLAS | 20.3 | 8 TeV | $WW\to\ell\nu\ell\nu$ | $p_{T\ell}^{(1)}$ |  [38], Fig. 11
CMS | 19.4 | 8 TeV | $WW\to\ell\nu\ell\nu$ | $m_{\ell\ell}$ |  [39], Fig. 4
ATLAS | 20.3 | 8 TeV | $WZ\to\ell\nu\ell\ell$ | $p_{TZ}$ |  [40], Fig. 5
CMS | 19.6 | 8 TeV | $WZ\to\ell\nu\ell\ell$ | $p_{TZ}$ |  [41], Fig. 7
ATLAS | 13.3 | 13 TeV | $WZ\to\ell\nu\ell\ell$ | $m_{WZ}$ |  [42], Fig. 3
Table 1: Data used to extract the current LHC bounds.

To perform the fit, we calculate the BSM cross sections at tree level with
MadGraph5 [45], while using FeynRules 2.0 [36] to generate the .ufo file for
the BSMC model [35]. This procedure gives the cross section in terms of the
seven BSM parameters $\delta g_{L,R}^{Zu,d}$ and $\delta
g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$. We perform a simulation to
get the cross section for each bin for every differential distribution shown
in Table 1, and then perform the cuts as described by the experimental
collaborations in each case. 777In some cases, like $WW\to\nu\ell\nu\ell$, the
cuts performed by the experiments for some sub-chanels are performed using a
Boosted Decision Tree and not just a cut and count approach. In this case we
only generate the subchannel for which we can easily reproduce the cuts, i.e.
$WW\to\nu_{e}e\nu_{\mu}\mu$ and then fit to the total combination assuming
that it does not depend on the lepton flavour. To get the BSM cross section,
we have generated for each bin several simulations corresponding to different
values of the BSM coefficients and then we have fitted them to a general
quadratic polynomial of the seven BSM coefficients $\delta g_{L,R}^{Zu,d}$ and
$\delta g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$ which we
schematically call $\delta_{i}$. In other words, we write 888When simulating
the BSM cross sections, we modify the four $Z\bar{q}q$ couplings $\delta
g_{L}^{Zu}$, $\delta g_{L}^{Zd}$, $\delta g_{R}^{Zu}$, $\delta g_{R}^{Zd}$,
for all the quark generations at the same time, as one would do in the FU
case, see Eq. (41). Nonetheless, due to the proton’s PDF, the contribution of
the light quarks $u,d$ is more than a factor ten greater than the one of
$c,s$, so one can safely assume that the modifications of $Zqq$ for second and
third generation give negligible contributions to diboson production. We
expect that the results we get for the diboson fit on the $Z\bar{q}q$
couplings for the FU case also apply for the $Z\bar{q}q$ couplings for the
first two generations of the MFV case, since in the MFV case $[\delta
g_{L,R}^{Zu,d}]_{11}\simeq[\delta g_{L,R}^{Zu,d}]_{22}$.

$\sigma_{SM+BSM}(\delta_{1},...,\delta_{n})=\sigma_{SM}+a_{i}\,\delta_{i}+b_{ij}\,\delta_{i}\delta_{j}\,,$
(11)

where the indices $i,j$ go from $i,j=1,...,7$, where $\sigma_{SM}$ corresponds
to the SM contribution, and $a_{i}$ and $b_{ij}$ are numerical coefficients
that characterize the BSM contribution which we determine by varying the BSM
parameters $\delta_{i},\delta_{j}$ in the MadGraph5 simulation. For $n$ number
of BSM parameters one has $n$ independent $a_{i}$ coefficients and
$n+n(n-1)/2$ independent $b_{ij}$ coefficients. So, for $n=7$, $35$
coefficients in total have to be fitted to obtain the full expression (11). We
then built the ratio $\delta\mu$ defined as

$\mu(\vec{\delta})\,=\,\frac{\sigma_{SM+BSM}(\vec{\delta})}{\sigma_{SM}}=1+\frac{\sigma_{BSM}(\vec{\delta})}{\sigma_{SM}}=1+\delta\mu(\vec{\delta})\,.$
(12)

Currently, the fully leptonic $WW$ and $WZ$ cross sections have been computed
at NNLO in QCD taking into account both on-shell and off-shell contributions
[46, 47, 48] and at NLO in EW but only on-shell [49, 50]. If, in the fiducial
phase space considered, the effects of taking into account the NLO corrections
can be encapsulated by an overall $k$-factor, 999This would not be the case if
the LO amplitude is highly suppressed. This is actually what is happening for
the WZ production channel as emphasized in Ref. [12]: in the central region of
the detector the $\pm 0$ and $\pm\mp$ LO amplitudes exactly vanish. We thank
G. Panico for pointing this out to us. the higher order corrections will
mostly cancel in the ratios $\delta\mu$, and that is why we use them to
perform the global fit which, as mentioned, is done at LO. This might not hold
for transverse polarizations as a result of the non-interfering effects
pointed out in Ref. [9], and an analysis can be found in Ref. [10]. Finally,
we build a $\chi^{2}$ function

$\chi^{2}=\sum_{I\in\text{channels}}\,\sum_{i\in\text{bins}}\frac{(\tilde{\sigma}_{SM}^{\textit{bkg}}+\mu\,\tilde{\sigma}_{SM}^{\textit{signal}}-\sigma_{\textit{measured}})^{2}_{I,i}}{(\Delta_{\textit{syst}})_{I,i}^{2}+(\Delta_{\textit{stat}})_{I,i}^{2}},\,$
(13)

where the first sum runs through all the channels under study, and the second
sum runs over each bin for the chosen differential distribution,
$\sigma_{measured}$ is the measured cross section including signal and
background, $\tilde{\sigma}_{SM}^{bkg}$ and
$\tilde{\sigma}_{SM}^{\textit{signal}}$ correspond to the simulated cross
sections for the signal and background done by the experimental
collaborations, $\Delta_{\textit{syst}}$ is the theoretical uncertainty given
by the experimental collaborations on the predicted SM cross sections,
$\tilde{\sigma}_{SM}^{\textit{bkg}}$ and
$\tilde{\sigma}_{SM}^{\textit{signal}}$, and finally $\Delta_{\textit{stat}}$
is the statistical error. When needed, we multiply and divide Eq. (13) by the
integrated luminosity squared and compute the $\chi^{2}$ function using the
number of events shown in the figures referred to in Table 1.

From the correlation matrices, central values and errors given in Ref. [27]
101010We thank the authors of Ref. [27] for providing the Mathematica code
with all the aforementioned quantities that had more precision than in the
paper, and allowed to get a more reliable $\chi^{2}$ for LEP-1., we build a
$\chi^{2}$ function for the LEP-1 measurements at the $Z$-pole. To perform the
global fits to get the aTGC bounds, we combine the two $\chi^{2}$ for diboson
at the LHC and LEP-1 as $\chi^{2}=\chi^{2}_{LHC}+\chi^{2}_{LEP-1}$.

###  3.2 Correlations among the Higgs basis parameters

When performing a $\chi^{2}$ fit, in the Gaussian limit, one can easily find
the correlation between two parameters by looking at the entries of the
correlation matrix. In our case, given that the $\chi^{2}$ function is not
Gaussian due to the non-negligible size of the $d=6$ BSM quadratic amplitudes,
we cannot easily extract a correlation matrix. Therefore, to get a sense of
the correlations among the different BSM coefficients, we perform a global fit
and look at the two dimensional plots for each pair of coefficients profiling
over all others. We show all these correlations in Appendix A.

As an example of the correlations among the different parameters, in the
center of Fig. -898, we show the projection of the $\chi^{2}$ function onto
the two dimensional plane $\left(\delta\kappa_{\gamma},\,\delta
g_{R}^{Zu}\right)$. The least constrained direction in this plot follows the
slope given by the combination appearing in the amplitude $\mathcal{M}(RR;00)$
of Eq. (6). The high energy flat direction is about ten times less constrained
than orthogonal direction, in agreement with the naive estimate made in
section 2.1.

The large correlation shown in the center of Fig. -898 makes $\delta
g_{R}^{Zu}$ and $\delta\kappa_{\gamma}$ very sensitive to each other. For
reference, we show in horizontal blue dashed lines the allowed 95% CL bounds
set by LEP-1 on $\delta g_{R}^{Zu}$ and in vertical the 95% CL bounds set by
LEP-2 on $\delta\kappa_{\gamma}$.

From this plot one can intuitively see that if $\delta g_{R}^{Zu}$ is not set
to zero but can vary within the range allowed by LEP-1, one may modify the
bounds on $\delta\kappa_{\gamma}$ in a non-negligible way. This indicates that
the bounds on the aTGCs should include the $\delta V\bar{q}q$ deformations if
a FU or a MFV scenario is assumed. Also, one can see that the assumptions on
$\delta\kappa_{\gamma}$ will have a large impact on the sensitivity of diboson
production to $\delta g_{R}^{Zu}$. We see that the sensitivity of diboson to
the different parameters is ultimately limited by the correlations, making a
global combination crucial.

Figure -898: $68\%$ (dark shaded) and $95\%$ (light shaded) CL regions using
the LHC diboson data reported in Table 1. Center: fit to
$\delta\kappa_{\gamma}$ and $\delta g_{R}^{Zu}$ profiling over all other five
parameters. The line shows the expected flat direction in the
$\hat{s}\rightarrow\infty$ limit that can be deduced from Eq. (6). Left
(Right): in yellow the fit to aTGC (vertex corrections) marginalising over all
other parameters, and in red (pink) the fit when the four $\delta V\bar{q}q$
(three aTGC) are set to zero.

Fortunately, in a broad class of models, the parameter $\delta\kappa_{\gamma}$
is expected to be generated only via loops, and, parametrically smaller than
the other parameters, it can be neglected when setting constraints. The same
holds true for $\lambda_{\gamma}$ which is also typically loop suppressed.
This is because both $\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$ modify the
magnetic moment and electric quadrupole moment of the $W$ which are only
generated at one loop in minimally coupled theories [33, 52]. Because of the
large correlations, setting them to zero can greatly increase the accuracy of
the fit to the various $\delta V\bar{q}q$.

###  3.3  $\delta V\bar{q}q$: LHC bounds vs LEP-1 constraints

In Fig. -897, we show the allowed 95% CL regions for the BSM coefficients
$\delta g_{L}^{Zu}$, $\delta g_{R}^{Zu}$, $\delta g_{L}^{Zd}$, $\delta
g_{R}^{Zd}$ defined in Eqs. (2) and (3), assuming i) that the aTGC are not
negligible (yellow), ii) that $\lambda_{\gamma}=\delta\kappa_{\gamma}=0$
(blue) and iii) that $\lambda_{\gamma}=\delta\kappa_{\gamma}=\delta g_{1z}=0$
(pink). In gray we show the bounds extracted from the LEP-1 fit of Ref. [27],
assuming that the EFT obeys either a MFV (light gray) or a FU (dark gray)
flavour structure. To avoid confusion, we remind that when extracting the
diboson bounds, we do not differentiate the cases of MFV and FU since diboson
production is mostly insensitive to possible differences between the light
generations and the third generation that could appear in the MFV case; the
only difference is a matter of interpretation, i.e. if one assumes FU the
diboson bounds on the $Z\bar{q}q$ anomalous couplings apply to all the three
quark generations, while if one assumes MFV they only apply to $u,\,d,\,c$ and
$s$ quarks.

We find that even for the most general case that includes all the seven BSM
parameters (yellow), the diboson bounds for the down-type couplings are
already competitive with those from LEP-1 one under the MFV scheme. The LHC
bound on $\delta g_{R}^{Zd}$ is better than the LEP-1 under the MFV hypothesis
and it remains competitive under the FU assumption. On the contrary, for the
up type quarks, we find that the LHC bounds are still significantly worse than
those from LEP-1, even under the MFV assumption.

Assuming that $\lambda_{\gamma}=\delta\kappa_{\gamma}=0$ (blue), we find a big
improvement on the diboson fit with respect to the seven parameter fit
(yellow). The most striking difference being that for the up-type quark
couplings, $\delta g_{L}^{Zu}$ and $\delta g_{R}^{Zu}$, the diboson bounds
become of the same order of magnitude as those from LEP-1; from these two
couplings, it is $\delta g_{R}^{Zu}$ that benefits the most from setting
$\lambda_{\gamma}=\delta\kappa_{\gamma}=0$. Notice that the improvement to
$\delta g_{R}^{Zu}$ is due to setting $\delta\kappa_{\gamma}=0$, since as
shown in Fig. -898 they are strongly correlated. On the other hand $\delta
g_{R}^{Zu}$ is insensitive to $\lambda_{\gamma}$. For the down type couplings,
we also find an improvement of about a factor two when setting the two aTGC to
zero. With these improvements, the current LHC diboson data set constraints on
$\delta g_{R}^{Zd}$ that are of same order as those derived from LEP-1 data in
a FU setup. For MFV scenarios, the LHC bounds significantly outperform the
LEP-1 ones.

Figure -897: 95% CL regions for the anomalous couplings between the light
quarks and the electroweak bosons. In light (dark) gray, the LEP-1 constraints
assuming MFV (FU). In yellow, the diboson bounds after profiling over the
remaining five parameters. In blue (pink) the same but setting
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$
($\delta\kappa_{\gamma}=\lambda_{\gamma}=\delta g_{1z}=0$).

In pink, we report the constraints for scenarios in which all the three aTGC
are negligible compared to $\delta V\bar{q}q$. Actually, letting $\delta
g_{1z}$ float or not does not significantly change the conclusion: we see that
the left handed couplings get a significant improvement with respect to the
blue region, while the right handed ones are almost insensitive to this extra
assumption on $\delta g_{1z}$. This, again, can be understood from the
“correlation matrix” of Fig. -888 that shows that $\delta g_{1z}$ is mostly
correlated with the left handed couplings.

Note that the correlations among the left and right couplings from LEP-1
measurement are not aligned with the correlation appearing in $pp\to WV$ data,
which gives some synergetic value to the combination of the two sets of data.
This can be seen for example in Table 3 that gives the individual constraints
from diboson and LEP-1 and their combination when
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. Notice also that while the LEP-1
data for down quarks has a two sigma excess (driven by the $Zb\bar{b}$
asymmetry) in an analysis in a FU context, the LHC diboson data presents a two
sigma excess as well, but in the opposite direction. So the combination
alleviates the tension with the SM.

One should remember that the bounds from $pp\to WV$ in Fig. -897 only
constrain BSM theories where the new particles are above few TeV (see the
discussion on the validity of EFT analysis in section 5), while those from
LEP-1 apply to theories where the new particles can be as light as
$\gtrsim{\cal O}(100)$ GeV.

To conclude this section, we note that in the fits of the $pp\to WV$ data, the
quadratic amplitudes appear to be non-negligible, modifying the constraints by
a factor $\sim 1.5-2$ when $\delta\kappa_{\gamma}$ is neglected, and by a
larger factor when $\delta\kappa_{\gamma}$ is taken into account, as a result
of the correlations identified earlier. We comment on what it means for the
EFT interpretation and possible BSM models in section 5.

###  3.4 LHC bounds on aTGC and interplay with $\delta V\bar{q}q$

Figure -896: 95% CL regions for the aTGCs. The red curves show the bounds from
a three parameter fit of the current LHC diboson data, imposing $\delta
V\bar{q}q=0$ and profiling over the aTGC not shown on the plots. The green and
blue curves show the bounds from a seven parameter global fit using the
current LHC diboson data as well as the LEP-1 data under the MFV and FU
assumptions respectively; the parameters not shown on the plots are profiled
over.

Figure -896 presents the 95% CL regions for the three aTGC parametrized by
$\delta g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$. In red, we show a
fit to the three aTGC setting $\delta g_{L}^{Zu}=\delta g_{R}^{Zu}=\delta
g_{L}^{Zd}=\delta g_{R}^{Zd}=0$ and profiling over the one aTGC not appearing
in the plot. In this case we only use the LHC data from Table 1. In dashed
green and dotted blue, we make a fit to the seven BSM parameters, the three
aTGC $\delta g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$ and the four
$\delta g_{L,R}^{Zu,d}$, and profile over those not appearing in the plot; in
this case we use $\chi^{2}=\chi^{2}_{diboson}+\chi^{2}_{\textit{LEP-1}}$,
assuming FU (dashed green) and MFV (dotted blue).

From Fig. -896, we see that the effect of not neglecting the $\delta
V\bar{q}q$ is the largest in the $(\delta\kappa_{\gamma},\delta g_{1z})$
plane, where the constrained area in parameter space varies around $50\%$ from
one assumption to the other. This points to a large correlation between
$\delta\kappa_{\gamma}$ and $\delta g_{1z}$ on the $\delta V\bar{q}q$
parameters, which is to be expected since they appear in the same high energy
amplitudes as seen in Eqs. (6) and (7). The determination of
$\lambda_{\gamma}$ is insensitive to the different assumptions, as expected
from the fact that it is the only parameter appearing in the amplitudes that
grow with $\hat{s}$ and have final polarizations $\pm\pm$.

Given that in many BSM models $\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$
are assumed to be loop induced and therefore parametrically smaller than
$\delta g_{1z}$, we also study the effect of profiling over $\delta V\bar{q}q$
when $\delta g_{1z}$ is the only aTGC modifying the diboson production. Since
the global fit is non-Gaussian, this particular case with
$\delta\kappa_{\gamma}=\lambda_{z}=0$ cannot be obtained simply from the
general case. On the left plot of Fig. -895, we show in solid black the one
parameter exclusive fit to $\delta g_{1z}$, setting all the other parameters
to zero. In dashed green and dotted blue, we allow $\delta g_{L}^{Zu},\,\delta
g_{R}^{Zu},\,\delta g_{L}^{Zd},\,\delta g_{R}^{Zd}$ to be different than zero
and perform a global fit. On the right plot of Fig. -895 we perform the
analysis separating the $WW$ and $WZ$ channels, and find that currently the
$WZ$ channels dominate the total $\Delta\chi^{2}$.

Figure -895:  Constraints on $\delta g_{1z}$ assuming $\delta\kappa_{\gamma}$
and $\lambda_{\gamma}$ to be loop suppressed, i.e. negligible. In solid,
exclusive fits where only $\delta g_{1z}$ is taken into account. In dashed and
dotted, fits profiling over the fermion-electroweak boson vertices under the
two different MFV and FU assumptions respectively. Left: $\Delta\chi^{2}$
combining all the channels shown in Table 1. Right: Showing the
$\Delta\chi^{2}$ for the $WW$ and $WZ$ channels separately.

Similarly to Fig. -896, Fig. -895 tells us that profiling $\delta V\bar{q}q$
in the fit changes the current constraints on $\delta g_{1z}$ by a factor of
about $25\%$. Also, we find that once the $\delta V\bar{q}q$ are introduced,
the FU and MFV assumptions which modify $\chi^{2}_{\textit{LEP-1}}$ yield
qualitatively similar size effects to $\delta g_{1z}$ but still with at least
10% differences between the two.

In this section, we have presented an analysis using all the current leptonic
diboson data to set constraints to both aTGCs and $\delta V\bar{q}q$ vertices,
and study the correlations under different flavour schemes. The LEP-1
constraints for $\delta V\bar{q}q$ in Fig. -897 could lead to the conclusion
that for a MFV setup larger deviations could be obtained. However, this is not
the case because the cross correlations among LEP-1 and LHC diboson data make
the global fit more constraining than both sets of data alone. The largest
correlation in the fit appears to be between $\delta\kappa_{\gamma}$ and
$\delta g_{R}^{Zu}$ as established in section 3.2.

##  4 Projected bounds for HL-LHC

###  4.1 Data used and assumptions for HL-LHC

To estimate the bounds at HL-LHC, as a first step and for simplicity, we
simulated the channels $pp\to WW\to\nu\ell\nu\ell$ and $pp\to
WZ\to\ell\nu\ell\ell$. We build a $\chi^{2}$ function with the same form as in
Eq. (13) and we inject the SM signal, i.e., we assume that the measured number
of events will be the same as in the SM prediction, so that
$\sigma_{measured}=\tilde{\sigma}_{SM}^{bkg}+\tilde{\sigma}_{SM}^{\textit{signal}}$
defined after Eq. (13). Therefore the $\chi^{2}$ can be written as:

$\chi^{2}=\sum_{I\in\text{channels}}\,\sum_{i\in\text{bins}}\frac{[\,\delta\mu(\vec{\delta}\,)]^{2}}{(\delta_{syst})_{I,i}^{2}+(\delta_{stat})_{I,i}^{2}}\,$
(14)

where we define $\delta_{syst}=\sqrt{b+({\cal L}\,\Delta_{syst})^{2}}/s$, with
$\Delta_{syst}$ being the absolute systematics error in the cross section,
${\cal L}$ being the integrated luminosity, and $\delta_{stat}=1/\sqrt{s}$.
$s$ and $b$ stand for the number of simulated SM signal and background events,
and $\delta\mu(\vec{\delta}\,)$ is defined in Eq. (12). As usual, events with
misidentified particles, such as misidentified leptons in processes with
$W+\text{jets}$ or top production (see e.g. section 5 of Ref. [39]), are
included within the background.

There has been no extensive study of the systematic uncertainties and the
expected background for $pp\to WV$, especially in the high energy bins. A 5%
of systematic uncertainties is claimed to be possible in Ref. [12] in the
fully leptonic $WZ$ channel within the fiducial region used for their
analysis, and is used as a benchmark in Ref. [23] for the semileptonic $WV$
and $Wh$ channels. In Ref. [4], instead, it is claimed that this accuracy can
only be reached by measuring ratios of cross sections. We take a pragmatic
approach and consider two scenarios for the uncertainties at HL-LHC: a
pessimistic one where $\delta_{syst}=30\%$ is assumed for all the bins, which
corresponds to an extrapolation of the uncertainty in the overflow bins of the
experimental analysis, and a more aggressive scenario where one assumes
$\delta_{syst}=5\%$ for all bins.

For the $WW$ channel, we consider the $m_{\ell\ell}$ distribution and for the
$WZ$ channel, we consider the $m_{T}^{WZ}$ distribution. In both cases we have
chosen the observables and cuts followed by the experimental collaborations in
[39, 42]. The binning used in our analysis also follows the experimental
collaborations for low transverse masses, while we add more bins at higher
transverse masses to increase the sensitivity. This corresponds to the
following binning,

$\displaystyle m_{T}^{WZ}$ $\displaystyle\in$
$\displaystyle\\{140,180,250,450,600,750,900,1100\\}\ \text{GeV}\,,$
$\displaystyle m_{\ell\ell}$ $\displaystyle\in$
$\displaystyle\\{50,125,200,300,500,700,900,1100,1300,1500,1700\\}\
\text{GeV}\,,$ (15)

where in each case the overflow bin is chosen to contain at least ten events.
As a small cross check, we compared our estimated bounds on the aTGC at HL-LHC
with 3 ab-1, shown in red in Fig. -893, with those in Fig. 3 of Ref. [53].
There the channels $W\gamma\to\ell\nu\gamma$ and $WZ\to\ell\nu\ell\ell$ were
considered and bounds on the aTGC were derived for a run at 14 TeV with a
total accumulated luminosity of up to 1 ab-1. Our bounds, assuming
$\delta_{syst}=5\%$ in the leptonic $WW$ and $WZ$ channels, turn out to be
more conservative than those in Ref. [53] but overall of the same order. So,
our simple assumptions are in line with the existing literature and should
give a reliable and conservative estimate of the HL-LHC reach. Note that there
are several ways to improve the diboson analysis: i) the semileptonic channels
can be considered on top of the purely leptonic ones, ii) more refined
observables like those presented in Refs. [12, 9, 11] can be studied.
Therefore even with $\delta_{syst}=5\%$, our estimates on the diboson reach at
HL-LHC are probably on the conservative side. To compare the traditional
experimental analysis with new proposals, in section 5.4 we compare the HL-LHC
reach of leptonic $WZ$ estimated in Ref. [12] with our combination of the
leptonic $WW$ and $WZ$ using the $m_{\ell\ell}$ and $p_{T}^{Z}$ differential
distributions.

###  4.2 HL-LHC projections on $\delta V\bar{q}q$ vs LEP-1

Figure -894: Estimated 95% CL bounds at HL-LHC on the anomalous couplings
between the light quarks and the electroweak bosons. In yellow, diboson bounds
after profiling over the remaining five parameters. In blue (pink), same but
setting also $\delta\kappa_{\gamma}=\lambda_{\gamma}=0$
($\delta\kappa_{\gamma}=\lambda_{\gamma}=\delta g_{1z}=0$). Solid and dashed
stand for an assumed $\delta_{syst}=5\%$ and $\delta_{syst}=30\%$
respectively. Light (dark) gray regions correspond to the LEP-1 bounds
assuming MFV (FU).

Figure -894 shows the allowed 95% CL regions for $\delta g_{L}^{Zu}$, $\delta
g_{R}^{Zu}$, $\delta g_{L}^{Zd}$, $\delta g_{R}^{Zd}$ in the three different
scenarios: i) the three aTGC, $\lambda_{\gamma},\delta\kappa_{\gamma},\delta
g_{1z}$, are kept as floating parameters in the fit (yellow), ii)
$\lambda_{\gamma}$ and $\delta\kappa_{\gamma}$ are set to zero (blue), and
iii) the three aTGC are set to zero (pink). A total accumulated luminosity of
3 ab-1 is assumed. In order to appreciate the improvement compared to LEP, the
gray regions report the bounds extracted from the LEP-1 data under the MFV
(light gray) and FU (dark gray) assumptions. Clearly, for low enough
systematics, HL-LHC will surpass the LEP-1 bounds for any new physics scenario
with a built in MFV structure that does not generate anomalously large aTGC,
i.e. scenarios for which $\delta\kappa_{\gamma}=\lambda_{\gamma}=0$ (blue) is
a good approximation. Under the FU assumption, the HL-LHC bounds on $\delta
g_{R}^{Zu}$ and $\delta g_{R}^{Zd}$ vastly surpass the LEP-1 bounds whenever
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$, while the bounds on $\delta
g_{L}^{Zu}$ and $\delta g_{L}^{Zd}$ are only slightly better. In any case, it
should be noted that the blue and pink bounds improve by one order of
magnitude at HL-LHC compared to the current bounds. As long as the systematics
remain low enough, the seven parameter FU fit also improves by about a factor
three the bounds for all the $\delta V\bar{q}q$ with respect to the current
bounds shown in Fig. -897. The seven parameter FU fit equals or surpasses the
LEP-1 constraints for $\delta g_{L}^{Zd},\,\delta g_{L}^{Zu}$ and $\delta
g_{R}^{Zd}$. On the other hand, with higher systematic uncertainties,
$\delta_{syst}=30\%$, the improvement from the seven parameter and five
parameter fits with respect to the current constraints will be limited and
mostly concern the right handed couplings. Only for $\delta g_{R}^{Zd}$, the
HL-LHC will show an improvement over LEP-1 in all the cases, both for MFV and
FU structures.

###  4.3 HL-LHC projections on aTGC and interplay with $\delta V\bar{q}q$

Figure -893 shows the allowed 95% CL regions for the three aTGC parametrized
by $\delta g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$. In red, we show
a fit to the three aTGC setting $\delta g_{L}^{Zu}=\delta g_{R}^{Zu}=\delta
g_{L}^{Zd}=\delta g_{R}^{Zd}=0$ and profiling over the one aTGC not appearing
in the plot. In green, we make a fit to the seven BSM parameters, namely the
three aTGC $\delta g_{1z},\,\delta\kappa_{\gamma},\,\lambda_{\gamma}$ and the
four $\delta g_{L,R}^{Zu,d}$ and we profile over those not appearing in the
plot. We use the HL-LHC projections to build $\chi^{2}_{diboson}$ while the
$\chi^{2}_{\textit{LEP-1}}$ is built from the global fits performed in Ref.
[27]. We find that at HL-LHC the differences between assuming MFV or FU for
$\chi^{2}_{\textit{LEP-1}}$ are negligible when performing a combined global
fit of LEP-1 and LHC data. For this reason in this section we only present
results with the FU hypothesis for the LEP-1 fit.

Figure -893: Estimated 95% CL bounds on the aTGC at HL-LHC. Solid and dashed
stand for $\delta_{syst}=5\%$ and $\delta_{syst}=30\%$ respectively. The
bounds on $\delta\kappa_{\gamma}$, $\delta g_{1z}$ change by a factor two to
three between the 3-parameter and the 7-parameter fits, while the bound on
$\lambda_{\gamma}$ remains unaffected. The aTGC bounds in 3-parameter fit
agrees well with the ones obtained for universal theories, see Fig. -883.

At HL-LHC, the aTGC bounds shown in Fig. -893 are qualitatively similar to
those of Fig. -896 obtained with the current data. The main difference between
the two is that the features found with the current data regarding the impact
of $\delta V\bar{q}q$ are accentuated at HL-LHC. This is particularly true for
$\delta\kappa_{\lambda}$ and $\delta g_{1z}$: the bounds on the
$\delta\kappa_{\gamma}$, $\delta g_{1z}$ vary by more than $100\%$ if instead
of setting $\delta V\bar{q}q=0$ they are included in a global fit combining
the LEP-1 data in the context of FU or MFV scenarios. On the other hand
$\lambda_{\gamma}$ will remain mostly unaffected, as anticipated from Eqs. (6)
and (7).

Figure -892 shows the 95%CL bound on $\delta g_{1z}$ when setting
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$ as a function of the assumed
systematic uncertainty. Two cases are considered: i) all deviations in the
light quark vertices are neglected and set to zero, and ii) the diboson data
are combined with the LEP-1 data and the light quark vertices are profiled
over. The bound on $\delta g_{1z}$ is rather robust and does not show a strong
dependence on the assumed systematic uncertainty, changing by a factor two
between when the systematics vary from 0% to 50% (the statistical uncertainty
is of course kept). The HL-LHC bound will be of the order of 0.1%, an order of
magnitude better than the current existing bound. And further improvement can
be anticipated, e.g. by relying on the new analyses proposed in Ref. [12].

Figure -892: 95 %CL bound at HL-LHC on $\delta g_{1z}$ as a function of the
assumed systematic uncertainty for the case where the light quark vertices are
neglected and for the case where we take them into account and combine the fit
with LEP-1 data. For comparison, in light gray we show the LEP-2 constraints,
rescaled by a factor 0.1. Left: $\Delta\chi^{2}$ combining the projections for
$WW$ and $WZ$. Right: Showing the projected $\Delta\chi^{2}$ for the $WW$ and
$WZ$ channels separately.

##  5 Interpretation of the constraints

In this work we have performed a global analysis of the diboson data at the
LHC and inferred bounds on aTGCs as well as on anomalous couplings of the
quarks to the EW gauge bosons. We found that in some cases these bounds
surpass the LEP-1 and LEP-2 bounds. Nonetheless, it is important to stress
that this is only so for certain regions of the parameter space. As in any EFT
analysis, the constraints on the Wilson coefficients are only valid when the
characteristic energy of the processes remains smaller than the masses of the
new particles. Furthermore, both for the current LHC data and also for the HL-
LHC ones, the quadratic terms of the BSM contributions to the diboson
production cross section play a non-negligible role in settings bounds on the
Wilson coefficients. In that situation, further restrictions on the parameter
space follow to ensure that the interference between the SM amplitude and the
dimension-8 operators, formally of the same order as the square of the
dimension-6 operator contributions, remains sub-dominant [17, 18, 9, 19, 15,
20, 21, 22, 5, 6, 11, 7, 10, 12]. We comment on these two limitations, in the
following, and also see how they appear in a concrete toy model with vector
triplets.

###  5.1 Quadratic BSM amplitudes

As already noted and extensively discussed in Refs. [17, 18, 9, 19, 15, 20,
21, 22, 5, 6, 11, 7, 10, 12], when setting bounds to the EFT coefficients, it
may happen that these bounds only constrain BSM amplitudes that are larger
than the SM one. This makes the quadratic dimension six BSM amplitudes to be
non negligible. To get a sense of which BSM theories can be studied only using
dimension six operators while neglecting those of dimension eight, it is
useful to schematically write the ratio of amplitudes between the EFT and the
SM. These estimates have already been discussed in Refs. [32, 17, 9], here we
only give a small review for convenience. Schematically, for the $WV$ channels
with longitudinally polarized gauge fields, the ratio of EFT and SM amplitudes
is given by

$\left|\frac{{\cal M_{\rm EFT}}}{{\cal M}_{\rm
SM}}\right|^{2}\,\sim\,1+\frac{c_{6}}{g_{SM}^{2}}\frac{E^{2}}{\Lambda^{2}}+\left(\frac{c_{6}^{2}}{g_{SM}^{4}}+\frac{c_{8}}{g_{SM}^{2}}\right)\frac{E^{4}}{\Lambda^{4}}+\cdots\,,$
(16)

where $c_{6},\,c_{8}$ represent the coefficients in front of the $d=6,8$
operators. When the quadratic terms dominate, the following condition has to
be fulfilled in order to be able to neglect the dimension-8 operators

$c_{6}^{2}\gg g_{SM}^{2}\,c_{8}\,.$ (17)

For simple power counting rules such that $c_{6}\sim c_{8}\sim g_{\star}^{2}$,
with $g_{\star}$ a charactestic coupling of the new physics degrees of
freedom, the EFT validity condition simply requires that the BSM coupling must
be larger than the SM one, $g_{\star}^{2}\gtrsim g_{SM}^{2}$, which is nothing
else than the condition that also ensures that the quadratic BSM pieces
dominate Eq. (16).

In the channels with mixed longitudinal and transverse polarizations, for
which the new physics amplitude only grows as $\sqrt{\hat{s}}$, the same
conclusion applies. The channel with transverse polarizations only is,
however, slightly different. In that case, the linear/interference terms at
the dimension-6 level is suppressed due to the necessity to go through a
helicity flip [32]:

$\left|\frac{{\cal M_{\rm EFT}}}{{\cal M}_{\rm
SM}}\right|^{2}\,\sim\,1+\frac{c_{6}}{g_{SM}^{2}}\frac{m_{W}^{2}}{E^{2}}\frac{E^{2}}{\Lambda^{2}}+\left(\frac{c_{6}^{2}}{g_{SM}^{4}}+\frac{c_{8}}{g_{SM}^{2}}\right)\frac{E^{4}}{\Lambda^{4}}+\cdots\,.$
(18)

And the quadratic pieces can dominate the linear terms for smaller values of
$g_{\star}$. One would then end up in a region of the parameter space where
the EFT analysis would not be valid since the dimension-8 operators cannot be
neglected. Recently, new observables have been proposed [9, 11] to resurrect
the interference and then circumvent this (in)validity issue.

###  5.2 Power countings and BSM interpretations

Assessing the consistency of EFT interpretation requires some assumptions on
the scaling of the Wilson coefficients of the higher dimensional operators. We
present here different power counting and selection rules which inspired the
particular choices of the BSM parameters kept in the fits presented in
sections 3 and 4. They correspond to specific dynamics for the new physics
above the weak scale. We follow the conventions of [25],

$\quad{\cal L}={\cal L}_{SM}+\sum_{d=6}\bar{c}_{i}\,{\cal
O}_{i}+\dots\quad\quad\text{with}\quad\quad\bar{c_{i}}=c_{i}\,v^{2}/\Lambda^{2}=\hat{c}_{i}\,m_{W}^{2}/m_{\star}^{2},$
(19)

where $[v]=[\Lambda]=\sqrt{\hbar}/L$, $[m_{W}]=[m_{\star}]=1/L$ and $c_{i}$
and $\hat{c}_{i}$ are dimensionless for all the operators except for ${\cal
O}_{3W}$ which has dimensions of $1/L^{4}$ hence $c_{3W}$ and $\hat{c}_{3W}$
have dimensions of $[g^{-2}]=\hbar$. 111111Notice that we use the notation
${\cal O}$ instead of $O$ in Eq. (19) with respect to the notation of [25].

  * •

For generic weakly coupled and renormalizable BSM gauge theories, with new
particles of spin at most equal to one and with interactions mediated by
operators of mass dimension smaller or equal to 4, one can check that, after
integrating the BSM sector the operators, defined in the SILH basis [25],
${\cal O}_{HB}$, ${\cal O}_{HW}$ and ${\cal O}_{3W}$ can only be generated at
one loop, while ${\cal O}_{B}$, ${\cal O}_{W}$, ${\cal O}_{2B}$, ${\cal
O}_{2W}$ can be generated at tree level. 121212See an example of this
procedure in [51].. Given that at dimension six

$\delta\kappa_{\gamma}\sim\bar{c}_{HB}+\bar{c}_{HW}\,,\hskip
28.45274pt\lambda_{\gamma}\sim g^{2}\,\bar{c}_{3W}\,,\hskip 28.45274pt\delta
g_{1z}\sim\bar{c}_{W}+\bar{c}_{2W}+\frac{g^{\prime
2}}{g^{2}}\,\bar{c}_{B}+\bar{c}_{HW}\ \dots\,\,,$ (20)

one expects $\delta\kappa_{\gamma},\lambda_{\gamma}$ to be loop suppressed
with respect to $\delta g_{1z}$ and therefore the leading effect is expected
in $\delta g_{1z}$.

As shown in previous sections, diboson production can also be affected by the
vertex corrections $\delta V\bar{q}q$. In weakly coupled gauge theories these
can also be generated at tree level, either by contributions from the bosonic
operators ${\cal O}_{B}$, ${\cal O}_{W}$, ${\cal O}_{2B}$, ${\cal O}_{2W}$ or
via ${\cal O}_{HQ}^{(1)}$, ${\cal O}_{HQ}^{(3)}$, ${\cal O}_{Hu}$, ${\cal
O}_{Hd}$, see for instance the model in section 5.4. Therefore, whether they
can be neglected in the diboson process or not depends on the specific details
of the UV theory. The flavour assumptions of the UV theory will also determine
the strength of the LEP-1 bounds.

  * •

Strongly-Interaction-Light-Higgs (SILH) models [52] address the hierarchy
problem by making the Higgs boson a pseudo Nambu–Goldstone boson associated
with the symmetry breaking of a global symmetry of a new strong sector. Hence,
the Higgs boson is a composite particle belonging to the strong sector. On the
other hand, the gauge bosons and fermions appearing in the SM are assumed to
be elementary and external to the strong sector and they acquire their masses
by mixing linearly to the resonances of the strong sector, a setup dubbed
partial-compositness. Assuming that the strong sector can be characterised by
one mass scale $m_{\star}$ and one coupling $g_{\star}$, it is possible to
estimate the size of the various EFT coefficients appearing after integrating
it out, see Ref. [52]. Focusing on the bosonic operators relevant to the aTGC
and the electroweak precision tests, their estimated size is found to be of
the order

$\displaystyle\hat{c}_{WH},\,\hat{c}_{HB},\sim\frac{g_{\star}^{2}}{16\pi^{2}}\,,\quad\hat{c}_{3W}\sim\frac{1}{16\pi^{2}},\quad\hat{c}_{2B}\sim\frac{g^{2}}{g_{\star}^{2}}\,,\quad\hat{c}_{2W}\sim\frac{g^{\prime
2}}{g_{\star}^{2}}\,,\quad\hat{c}_{B},\hat{c}_{W}\sim{\cal O}(1)\,,$ (21)

which yields the following power countings for the aTGC’s:

$\delta g_{1z}\sim\frac{m_{W}^{2}}{m_{\star}^{2}}\,,\hskip
56.9055pt\delta\kappa_{\gamma}\sim{g_{\star}^{2}\over
16\pi^{2}}\frac{m_{W}^{2}}{m_{\star}^{2}}\,,\hskip
56.9055pt\lambda_{\gamma}\sim{g_{SM}^{2}\over
16\pi^{2}}\frac{m_{W}^{2}}{m_{\star}^{2}}\,.$ (22)

Therefore for SILH-like models, $\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$
are parametrically suppressed by one loop with respect to $\delta g_{1z}$.

Let us now asses the size of the vertex corrections $\delta V\bar{q}q$ that
can also modify the diboson production. If the leading operators modifying the
$W,\,Z$ couplings to the quarks are bosonic, one ends up with

$\delta g_{L,R}^{Zu,d}\sim\frac{m_{W}^{2}}{m_{\star}^{2}}.$ (23)

which is comparable in size to the deviation expected for $\delta g_{1z}$.
However in this case the LEP-1 bounds are very constraining (see subsection on
universal theories in Appendix B) and the $\delta V\bar{q}q$ vertex
corrections can be neglected in our analysis of diboson production. On the
other hand, if one has non-bosonic operators modifying $\delta V\bar{q}q$,
then, the LEP-1 bounds can be relaxed such that these vertex corrections
affect the diboson production. This can happen when the light quarks have a
high degree of compositeness $\epsilon_{q}$ due to linear mixings of the form
$\epsilon_{q}m_{\star}\bar{q}\Psi$, where $q$ stands for either a quark double
or singlet and $\Psi$ a heavy resonance. In this case, it is possible to
generate sizeable and uncorrelated coefficients for the operators ${\cal
O}_{Hq}^{(3)}$, ${\cal O}_{Hq}^{(1)}$, ${\cal O}_{Hu}$, ${\cal O}_{Hd}$ which
yield vertex corrections of order:

$\delta
g_{L,R}^{Zu,d}\sim\epsilon_{q}\frac{g_{\star}^{2}}{g_{SM}^{2}}\,\frac{m_{W}^{2}}{m_{\star}^{2}}.$
(24)

as shown in the example of section 5.4. In this scenario, the LEP-1 constrains
are expected to be less stringent than in the universal case (when the EFT is
described only by bosonic operators), and therefore the vertex corrections
$\delta V\bar{q}q$ may affect diboson production at the LHC.

  * •

Strongly Coupled Multi-pole Interaction models [54] are hypothetic strongly
coupled UV theories where the fermions and gauge bosons appearing in the SM
can be resonances of a strong sector; nonetheless their EFT’s can exhibit
couplings at dimension four, $g_{SM}$, that are much smaller than the
characteristic coupling of the strong sector $g_{\star}\lesssim 4\pi$. On the
other hand, the strong coupling $g_{\star}$ only manifests via higher
derivative interactions coming from operators with $d>4$. The resulting power
counting for these theories, when the SM gauge bosons are part of the strong
sector, shows that $\lambda_{\gamma}$ and $\delta\kappa_{\gamma}$ can be
generated without the loop suppression. The three scenarios presented in Ref.
[54] Pure Remedios, Remedios+MCHM and Remedios+ISO(4) induce the following
$d=6$ operators with sizes of order

$\hat{c}_{3W}\sim\frac{g_{\star}}{g_{SM}^{3}}\,,\hskip
56.9055pt\hat{c}_{2B},\hat{c}_{2W}\sim{\cal O}(1)\,.$ (25)

where $g_{\star}\gtrsim g_{SM}$. On the other hand only the R+MCHM and
R+ISO(4) generate the operators ${\cal O}_{HB}$, ${\cal O}_{HW}$, ${\cal
O}_{W}$ and ${\cal O}_{B}$ with sizes of
$\hat{c}_{HB},\hat{c}_{W},\hat{c}_{B}\sim{\cal O}(1)\,,$ while $\hat{c}_{HW}$
is of order ${\cal O}(1)$ and $g_{\star}/g$ for R+MCHM and R+ISO(4)
respectively. These power counting imply that for all these models, i.e. Pure
Remedios, R+MCHM and R+ISO(4),

$\lambda_{\gamma}\sim\frac{g_{\star}}{g_{SM}}\frac{m_{W}^{2}}{m_{\star}^{2}}\,.$
(26)

On the other hand, for the Pure Remedios case, $\delta
g_{1z}\sim\frac{m_{W}^{2}}{m_{\star}^{2}}$ while $\delta\kappa_{\gamma}$ is
not generated at dimension six. For the other two cases one has that

$\delta
g_{1z}\sim\delta\kappa_{\gamma}\sim\frac{m_{W}^{2}}{m_{\star}^{2}}\quad(\text{{\it
R}+MCHM})\,,\hskip 28.45274pt\delta
g_{1z}\sim\delta\kappa_{\gamma}\sim\frac{g_{\star}}{g_{SM}}\frac{m_{W}^{2}}{m_{\star}^{2}}\quad(\text{{\it
R}+ISO(4)})\,.$ (27)

Hence, Remedios-type models show possible ways in which
$\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$ could be not loop suppressed,
giving further motivation for studying their bounds, while showing that
neglecting these two aTGC may not always be appropriate.

###  5.3 Energy limitation

An EFT has an intrinsic cutoff scale and for the analysis to be valid it
should not use any event with a characteristic scale above this cutoff. Given
that the center of mass energy of the interacting partons is not known at the
LHC, it may be impossible to know the center of mass energy of a given process
if the energy and momentum of the final states are not completely
reconstructed. This is the case of the leptonic processes for $pp\to WV$ with
one or two neutrinos in the final states. We set 3 TeV as the energy for which
the EFT stops to be valid following the analysis from Ref. [6] and checking
that we did not get any events in the Madgraph5 simulation above 3 TeV. One
could extend the EFT reach below the 3 TeV mark without changing the
experimental analysis by following the procedure explained in Ref. [17, 15,
55, 56]. This procedure is based on considering only the events with a
characteristic energy below a pre-determined cutoff scale $E_{cut}$. The
constraints obtained this way are, although not optimal since one is throwing
away the events above the cutoff, totally consistent with the EFT expansion.

To illustrate the effect of the energy limitation to stay within the validity
region of an EFT analysis, one can consider the projections of the $\delta
V\bar{q}q$ bounds onto the parameter space $(g_{\star},m_{\star})$ on models
whose dynamics follows the power counting discussed above.

  * •

For SILH-like models with elementary quarks, the scaling (23) naively leads to
a simple 68%CL lower bound on $m_{\star}$ independent of $g_{\star}$:

$m_{\star}>(500,900,1300)\,\textrm{GeV at LEP-1, LHC, HL-LHC respectively}$
(28)

(we used the constraints on $\delta g_{R}^{Zd}$ of Tables 3 and 5, under the
MFV assumption and setting the aTGC to zero). However, the bounds in Eq. (28)
for the LHC and HL-LHC are not reliable since they fall outside the regime of
validity of the EFT. Hence, only the LEP-1 bound can be trusted for these
types of models. Futhermore, as commented above, in this elementary quark SILH
scenario, there exist some correlations among the four $\delta V\bar{q}q$
couplings and a more meaningful bound on $(m_{\star},g_{\star})$ should take
into account these correlations. For instance, the 95%CL LEP-1 bound can be
obtained from the universal theory fit, see Eq. 43, leading to

$m_{\star}>2.3\,\textrm{TeV}.$ (29)

In deriving this constraint, we have used $\delta g_{R}^{Zd}\sim g^{\prime
2}\hat{S}/(3g^{2}-3g^{\prime 2})$ and assumed the scaling $\hat{S}\sim
m_{W}^{2}/m_{\star}^{2}$, where $\hat{S}$ is one of the oblique parameters
relevant for universal theories.

  * •

For the SILH-like models with composite quarks, the situation is different and
there the diboson channels at the LHC can be used to set reliable constraints
stronger than the ones derived at LEP. Indeed the scaling from Eq. (24)
imposes at 68%CL

$g_{\star}/m_{\star}<(1.3,0.7,0.4)\,{\rm TeV^{-1}}\,\textrm{at LEP-1, LHC, HL-
LHC respectively}.$ (30)

And this time, the validity constraint of $m_{\star}>3$ TeV implies that at
the LHC and HL-LHC, diboson data can reliably constrain theories with
$g_{\star}>2.1$ and $1.2$ respectively, i.e., with a characteristic coupling
slightly larger than the electroweak one.

###  5.4 A model with triplets: diboson reach vs other searches

In this section we put the previous results in a global perspective, assessing
the usefulness of diboson observables in a simple UV toy model where other
types of searches are also constraining the parameter space. Our motivation
stems from the fact that from an EFT point of view, non-universal corrections
to the light quark vertices come from operators of the type
$(\bar{f}\gamma_{\mu}f)(H^{\dagger}\overleftrightarrow{D}_{\mu}H)$, and in
general grounds one expects to also generate the operators
$(\bar{f}\gamma_{\mu}f)(\bar{f}\gamma_{\mu}f)$ and
$(H^{\dagger}D_{\mu}H)(H^{\dagger}D_{\mu}H)$, which affect dijet processes and
Higgs physics respectively. Considering a particular model allows one to
compare these different searches and appreciate their
complementarity.131313There are other possible BSM scenarios with signals in
diboson production but not in dijets, e.g., a model with vector-like quarks as
seen from Table 8 of Ref. [59]. It may be interesting to see what diboson
production can say about these types of scenarios.

We focus our attention to the general vector triplet models presented in Refs.
[57, 15, 58], which appear in various BSM scenarios, and can produce sizable
and non-universal deviations to $\delta V\bar{q}q$ for the light quarks. We
will see how the different searches are sensitive in the different limits of
the parameter space.

For generality, we give the expressions for a model with custodial symmetry
consisting on two vectorial resonances, $L_{\mu}$ and $R_{\mu}$, transforming
respectively as $(1,3,1)$ and $(1,1,3)$ under $SU(3)_{C}\otimes
SU(2)_{L}\otimes SU(2)_{R}$. At leading order, these resonances couple to the
SM currents as follows:

$\mathcal{L}_{\textit{int}}=L^{a}_{\mu}\left(\gamma_{H}J_{\mu}^{Ha}\,+\,\gamma_{V}J_{\mu}^{a}+\sum_{f}\gamma_{f}J_{\mu}^{fa}\right)\\\
+R^{0}_{\mu}\left(\delta_{H}J_{\mu}^{H}\,+\,\delta_{V}J_{\mu}+\sum_{f}\delta_{f}J_{\mu}^{f}\right)\,+\,\frac{1}{\sqrt{2}}(\delta_{H}R^{+}_{\mu}J_{\mu}^{-H}+h.c.)$
(31)

where the SM currents are given by

$\begin{split}&J_{\mu}^{Ha}=\frac{i}{2}H^{\dagger}\sigma^{a}\overleftrightarrow{D}_{\mu}H\,,\quad
J_{\mu}^{a}=D^{\nu}W^{a}_{\nu\mu}\,,\quad
J_{\mu}^{fa}=\bar{f}\gamma_{\mu}\sigma^{a}f\,,\quad
J_{\mu}^{H}=\frac{i}{2}H^{\dagger}\overleftrightarrow{D}_{\mu}H\,,\\\
&J_{\mu}=\partial^{\nu}B_{\nu\mu}\,,\quad
J_{\mu}^{f}=\bar{f}\gamma_{\mu}f\,,\quad
J_{\mu}^{H-}=\frac{i}{2}H^{T}\overleftrightarrow{D}_{\mu}H.\end{split}$ (32)

The simplified UV model is fully characterized by the 11 arbitrary parameters
$(\gamma_{H},\gamma_{V},\gamma_{f=Q_{L},\ell_{L}},$
$\delta_{H},\delta_{V},\delta_{f=Q_{L},u_{R},d_{R},\ell_{L},e_{R}})$. The
couplings to each fermion also carry flavour indices. In the following, we
will assume that they follow the MFV flavor scheme, with the two lighter
generations having roughly the same $\gamma_{f}$ and $\delta_{f}$ couplings,
and the third generation being different.

When both resonances have a mass $m_{\star}\gg m_{W}$, they can be integrated
out to generate higher-dimensional interactions among the SM particles. At
order $1/m_{\star}^{2}$, see Ref. [15], this yields

$\mathcal{L}_{\textit{tree}}^{(6)}\supset\bar{c}_{W}\,{\cal
O}_{W}+\bar{c}_{B}\,{\cal O}_{B}+\bar{c}_{2W}\,{\cal
O}_{2W}+\bar{c}_{2B}\,{\cal O}_{2B}+\bar{c}_{H}\,{\cal O}_{H}\\\\[5.69046pt]
+\sum_{f}(\bar{c}_{Hf}\,{\cal O}_{Hf}+\bar{c}^{(3)}_{Hf}\,{\cal
O}^{(3)}_{Hf})\,+\,\sum_{f,f^{\prime}}(\bar{c}_{ff^{\prime}}\,{\cal
O}_{ff^{\prime}}+\bar{c}^{(3)}_{ff^{\prime}}{\cal O}^{(3)}_{ff^{\prime}})\,,$
(33)

where the operators are defined as

$\displaystyle\displaystyle{\cal O}_{B}={ig^{\prime}\over
2m_{W}^{2}}\left(H^{\dagger}D_{\mu}H\right)\partial_{\nu}B_{\mu\nu}\,,\quad{\cal
O}_{W}={ig\over
2m_{W}^{2}}\left(H^{\dagger}\sigma^{i}D_{\mu}H\right)\partial_{\nu}W^{i}_{\mu\nu}\,,\quad{\cal
O}_{H}={1\over 2v^{2}}\left(\partial_{\mu}|H|^{2}\right)^{2}\,,$
$\displaystyle\displaystyle{\cal O}_{2B}={1\over
m_{W}^{2}}\left(\partial^{\mu}B_{\mu\nu}\right)^{2}\,,\quad{\cal
O}_{2W}={1\over
m_{W}^{2}}\left(\partial^{\mu}W^{i}_{\mu\nu}\right)^{2}\,,\quad{\cal
O}_{Hf}={i\over v^{2}}\bar{f}\gamma_{\mu}fH^{\dagger}D_{\mu}H\,,$ (34)
$\displaystyle\displaystyle{\cal O}_{Hf}^{(3)}={i\over
v^{2}}\bar{f}\sigma^{i}\gamma_{\mu}fH^{\dagger}\sigma^{i}D_{\mu}H\,,\quad{\cal
O}_{ff^{\prime}}={1\over
v^{2}}\left(\bar{f}\gamma_{\mu}f\right)^{2}\,,\quad{\cal
O}_{ff^{\prime}}^{(3)}={1\over
v^{2}}\left(\bar{f}\sigma^{i}\gamma_{\mu}f\right)^{2}\,.$

At tree-level, the matching between the UV model and its EFT description leads
to the following expression of the Wilson coefficients appearing in Eq. (34)

$\displaystyle\displaystyle\bar{c}_{B}=\frac{m_{W}^{2}}{m_{\star}^{2}}{\delta_{H}\delta_{V}\over
g^{\prime
2}}\,,\quad\bar{c}_{W}=\frac{m_{W}^{2}}{m_{\star}^{2}}{\gamma_{H}\gamma_{V}\over
g^{2}}\,,\quad\bar{c}_{H}=3\frac{m_{W}^{2}}{m_{\star}^{2}}\frac{\delta_{H}^{2}+\gamma_{H}^{2}}{g^{2}}\,,$
$\displaystyle\displaystyle\bar{c}_{2B}=\frac{m_{W}^{2}}{4m_{\star}^{2}}\frac{\delta_{V}^{2}}{g^{\prime
2}}\,,\quad\bar{c}_{2W}=\frac{m_{W}^{2}}{4m_{\star}^{2}}\frac{\gamma_{V}^{2}}{g^{2}}\,,\quad\bar{c}_{Hf}=\frac{m_{W}^{2}}{m_{\star}^{2}}\frac{2}{g^{2}}\left(-\delta_{H}\delta_{f}+\delta_{V}\delta_{f}\right)\,,$
(35)
$\displaystyle\displaystyle\bar{c}_{Hf}^{(3)}=\frac{m_{W}^{2}}{m_{\star}^{2}}\frac{2}{g^{2}}\left(-\gamma_{H}\gamma_{f}+\gamma_{V}\gamma_{f}\right)\,,\quad\bar{c}_{ff^{\prime}}=-\frac{m_{W}^{2}}{2m_{\star}^{2}}\frac{1}{g^{2}}\delta_{f}\delta_{f^{\prime}}\,,\quad\bar{c}^{(3)}_{ff^{\prime}}=-\frac{m_{W}^{2}}{2m_{\star}^{2}}\frac{1}{g^{2}}\gamma_{f}\gamma_{f^{\prime}}\,.$

In Eq. (33) the sums for $f,f^{\prime}$ can run over
$\\{Q_{L}^{i},u_{R}^{i},d_{R}^{i},\ell_{L}^{i},e_{R}^{i}\\}$ for the operators
${\cal O}_{Hf},\,{\cal O}_{ff^{\prime}}$, and over
$\\{Q_{L}^{i},\ell_{L}^{i}\\}$ for ${\cal O}^{(3)}_{Hf},\,{\cal
O}^{(3)}_{ff^{\prime}}$.

Among the seven parameters entering in fit to the diboson data, only five are
generated at tree-level: the aTGC $\delta g_{1z}$ and the four vertex
corrections $\delta g_{L,R}^{Zu,d}$. The two other aTGCs
$\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$ are generated at one loop by
the operators ${\cal O}_{HB},\,{\cal O}_{HW}$ and ${\cal O}_{3W}$ defined in
Table 97 of Ref. [26]. Specifically, we have141414We assume the CKM matrix to
be diagonal since the mixing effects are negligible in our analysis.,

$\displaystyle\delta g_{1z}=-\frac{g^{2}+g^{\prime 2}}{g^{2}-g^{\prime
2}}\left(\bar{c}_{W}+\bar{c}_{2W}+\frac{g^{\prime
2}}{g^{2}}(\bar{c}_{B}+\bar{c}_{2B})\right)\,,$ $\displaystyle\delta
g_{L}^{Zu}=\frac{1}{2}\left(-\bar{c}_{HQ}+\bar{c}_{HQ}^{(3)}+\bar{c}_{2W}+\bar{c}_{2B}\frac{g^{\prime
2}}{g^{2}}-\frac{2}{3}\frac{2g^{\prime 2}}{g^{2}-g^{\prime
2}}\left(\bar{c}_{2B}\frac{2g^{2}-g^{\prime
2}}{g^{2}}+\bar{c}_{2W}+\bar{c}_{B}+\bar{c}_{W}\right)\right)\,,$
$\displaystyle\delta
g_{L}^{Zd}=\frac{1}{2}\left(-\bar{c}_{HQ}-\bar{c}_{HQ}^{(3)}-\bar{c}_{2W}-\bar{c}_{2B}\frac{g^{\prime
2}}{g^{2}}+\frac{1}{3}\frac{2g^{\prime 2}}{g^{2}-g^{\prime
2}}\left(\bar{c}_{2B}\frac{2g^{2}-g^{\prime
2}}{g^{2}}+\bar{c}_{2W}+\bar{c}_{B}+\bar{c}_{W}\right)\right)\,,$
$\displaystyle\delta
g_{R}^{Zu}=\frac{1}{2}\left(-\bar{c}_{Hu}-\frac{2}{3}\frac{2g^{\prime
2}}{g^{2}-g^{\prime 2}}\left(\bar{c}_{2B}\frac{2g^{2}-g^{\prime
2}}{g^{2}}+\bar{c}_{2W}+\bar{c}_{B}+\bar{c}_{W}\right)\right)\,,$
$\displaystyle\delta
g_{R}^{Zd}=\frac{1}{2}\left(-\bar{c}_{Hd}+\frac{1}{3}\frac{2g^{\prime
2}}{g^{2}-g^{\prime 2}}\left(\bar{c}_{2B}\frac{2g^{2}-g^{\prime
2}}{g^{2}}+\bar{c}_{2W}+\bar{c}_{B}+\bar{c}_{W}\right)\right)\,.$ (36)

The operators ${\cal O}_{ff^{\prime}}$, ${\cal O}^{(3)}_{ff^{\prime}}$ and
${\cal O}_{H}$ do not contribute to diboson production but they modify dijet
and Higgs production which can then be used to set constraints on the
parameters $\gamma_{f},\,\delta_{f},\gamma_{H},\,\delta_{H}$ of the simplified
UV model.

#### General models with only $L_{\mu}$

In order to compare with experimental bounds and previous works, we will only
consider the scenario where one has only the $L_{\mu}$ resonance. We further
assume for simplicity that $\gamma_{V}\ll\gamma_{H},\gamma_{Q}$. This minimal
setup interpolates between the strongly and weakly coupled limits in the Higgs
and fermionic sectors.

Figure -891: 95% CL exclusion regions in the $(\gamma_{H},\gamma_{Q})$ plane
for various values of the resonance masses. In blue, the projected constraints
from diboson data at HL-LHC, using our projections (dark shade) or the refined
analysis strategy of Ref. [12]. The other constraints come from a recast of
the studies in Refs. [21, 60, 61, 62, 63] and are commented in detail in the
text.

Figure -891 shows, for fixed $m_{\star}$, the constraints on
$(\gamma_{H},\gamma_{Q})$ from various searches. The blue regions correspond
to constraints imposed by the future diboson measurements at HL-LHC, using our
projections (dark shade) or the refined the analysis strategy proposed in Ref.
[12] (light shade). For comparison, the LEP-1 bound taken from Ref. [27] under
the MFV flavour scheme is indicated by the dashed light blue line. For this
scenario, HL-LHC hardly competes with LEP-1. The systematics uncertainty in
diboson measurements would have to go significantly below 10% to overcome the
LEP-1 constraints.

For light resonance mass, $m_{\star}=5\,{\rm TeV}$ (left plot), the parameter
space is severely constrained by direct resonance searches in the leptonic
channel (yellow), hadronic channel (red) or diboson channel (dark red). The
bounds have been obtained by recasting the projections of Refs. [60, 61]). To
derive the bound from Drell–Yan searches, we set $\gamma_{\ell}=\gamma_{Q}$.
The bright red dotted line delineates the boundary between the regions in
which $L_{\mu}$ has a width smaller or larger than 20% its mass; this
separates the regions where the direct searches may stop being sensitive to
these resonances (at large $\gamma_{Q}$ and $\gamma_{H}$). Finally, the
sensitivity at HL-LHC in the Higgs coupling measurements also cuts off the
region with $\gamma_{H}>9$, corresponding to
$\xi=v^{2}g_{\star}^{2}/m_{\star}^{2}>0.08$ [60, 62, 63].

At higher resonance masses, $m_{\star}=7\,{\rm TeV}$ (center plot) and
$10\,{\rm TeV}$ (right plot), the direct resonance searches loose steam and
the diboson channels become more relevant in a larger portion of the parameter
space. Already for $m_{\star}=7\,{\rm TeV}$, the resonant dijet bound falls in
the region where $\Gamma/m_{\star}>20\%$, questioning its validity. The Higgs
and resonant diboson constraints are too weak to set any constrain at these
masses. For $m_{\star}=10\,{\rm TeV}$, the constraints from resonant Drell–Yan
searches fade away too.

Non-resonant dijet observables also impose severe constraints on the viable
parameter space of our simplified model, the dijet EFT lines in Fig. -891. We
used the results of Ref. [21] that puts bound on coefficient $Z$ of the
dimension-6 operator involving two gluon field strenghts

$\mathcal{L}\supset-\frac{Z}{2m_{W}^{2}}D_{\mu}G^{A\,\mu\nu}D^{\rho}G^{A}_{\rho\nu}\,.$
(37)

Using the equations of motion, this operator can be rewritten in terms four-
fermion operators.

The results presented in this section can be translated to the benchmark
models A and B suggested in Ref. [58]. The model A corresponds to a gauge
bosons from an extended gauge symmetry and it features
$\gamma_{H}\gamma_{f}\sim g^{4}/g_{\star}^{2}$ and $\gamma_{H}/\gamma_{f}\sim
1$. Figure -891 shows that this scenario is better probed via direct searches.
The model B corresponds to a resonances from a composite sector, and it
features $\gamma_{H}\gamma_{f}\sim 1$ and $\gamma_{H}/\gamma_{f}\sim
g_{\star}^{2}/g^{2}$, which projects in parameter space onto a line parallel
to the indirect diboson constraints in Fig. -891. The indirect probes can
bring information complementary to the direct constraints.

In conclusion, the diboson channels give interesting constraints in regions
where $\gamma_{Q}$ is small while $\gamma_{H}$ is large, which can be mapped
to composite models with heavy resonances strongly coupled to the Higgs boson
but weakly coupled to the light quarks. In this section, we only studied the
case with a left handed resonance $L_{\mu}$ in order to compare with the
direct experimental searches and with previous phenomenological works. We find
that, with an uncertainty $\delta_{syst}=10\%$, the $pp\to WW,WZ$ leptonic
channels at HL-LHC will access regions of the parameter space that remain
blind to other searches. Nonetheless, for this simplified scenario, LEP-1 is
still slightly better than our HL-LHC projections. It would be interesting to
study the case where more than one resonance is present, and therefore the
various $\delta V\bar{q}q$ are less correlated. We expect, as shown in
sections 3.3 and 4.2, that in these cases diboson production will be
significantly better than LEP-1 while being complementary to direct searches.

##  6 Summary and outlook

The high energies accessible at the LHC open the possibility not only to
directly produce new states, but also to enhance the sensitivity to new
physics out of direct reach with effects that are encoded in higher
dimensional operators involving the SM degrees of freedom. We offered a
detailed analysis of diboson processes at LHC, which provides an interesting
probe of some of these operators, in particular those that give rise to
effects growing with the characteristic energy scale of the underlying hard
process.

Due to the expected increased sensitivity in the analyses, we reiterated that
the interpretation of the diboson measurements in terms of anomalous triple
gauge couplings has to be reconsidered. In particular, the effects of
anomalous couplings among the light quarks and the electroweak bosons can no
longer be neglected a priori. On one hand, the current LHC diboson data
already set stronger constraints than LEP-1 on the anomalous couplings $\delta
V\bar{q}q$ for the down quark, at least under the hypothesis of MFV. On the
other hand, both in the MFV and FU hypotheses, the aTGC fit is found to be
only marginally stable under profiling over the $\delta V\bar{q}q$ vertex
corrections even when the LEP-1 constraints are imposed.

Figure -890: Current (left) and future (right) 95% CL constraints on the
anomalous vertices among light quarks and electroweak bosons. In gray, we show
LEP-1 results assuming MFV. In yellow we use LHC diboson data and perform a
global fit including the aTGCs. In blue, we only profile over $\delta g_{1z}$
and the vertex corrections. The thicker boxes combine LHC and LEP-1 data.

We did a simple estimate for the HL-LHC reach and found that the constraints
will improve by a factor two to three. The different flavour assumptions on
the vertices will have a seizable impact on the aTGC constraints. Quite
remarkably, the precision on light quark couplings at HL-LHC will
significantly surpass the LEP-1 constraints for both MFV and FU assumptions.
And, as shown in Fig. -892, the HL-LHC may be able to set bounds on $\delta
g_{1z}$ of the order of $0.1\%$ in both FU and MVF scenarios. On the contrary,
we checked that for universal theories which, as shown in Appendix B, depend
only on three aTGC and two $\delta V\bar{q}q$, the HL-LHC bounds are still far
from reaching the LEP-1 precision.

The left plot of Fig. -890 shows that the current leptonic diboson data can
already set bounds setting bounds on $\delta V\bar{q}q$ that are competitive
with LEP-1, and they can also improve the bound on $\delta g_{R}^{Zd}$. The
right plot shows that, by the end of HL-LHC, leptonic $pp\to WV$ can be very
competitive with the LEP-1 bounds or greatly surpass them if one assumes that
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. Focusing on the aTGCs, Fig. -889
tells that, even when $\delta\kappa_{\gamma}$ and $\lambda_{\gamma}$ are
neglected, $\delta g_{1z}$ is quite sensitive to the $\delta V\bar{q}q$
anomalous couplings: both with the current data and at HL-LHC, the $\delta
g_{1z}$ bounds varying by about 30% when $\delta V\bar{q}q$ are switch on and
off in the global fit.

Figure -889: Current (left) and future (right) 95% CL constraints on $\delta
g_{1z}$. In gray, LEP-2 results. In blue, constraint from LHC diboson data
alone. In green, fit to LHC diboson data including the anomalous $V\bar{q}q$
vertices and profiling over them with the LEP-1 MFV constraints. In the
projection of the HL-LHC bounds, a 10% systematic uncertainty in the channel
$pp\to WW,WZ$ is assumed.

We studied the interplay between the operators probed in diboson and the ones
probed in other searches, as dijets or Higgs physics. This interplay can be
intuitively understood by remembering that the operators affecting diboson
take the form $(\bar{f}\gamma_{\mu}f)(H^{\dagger}D_{\mu}H)$, and one can
expect that generically might be accompanied by operators like
$(\bar{f}\gamma_{\mu}f)^{2}$ and $(H^{\dagger}D_{\mu}H)^{2}$ as well. As a
concrete example, we presented a model in which all those deviations are
indeed induced, showing that measurements in diboson offer a complementarity
exploration of the parameter space. It would be interesting to see how the
direct and indirect bounds change for models with more resonances; we expect
that diboson will fare better compared to the other searches when considering
less simplified scenarios.

There are several interesting future directions. Focusing on the current
experimental searches, it would be interesting to study the semileptonic
channels, which might benefit from fat jet techniques [44], perhaps allowing
to reach higher invariant masses than the leptonic ones. Regarding new
searches, one could follow the steps advocated in Refs. [9, 12, 23, 11] and
find new ways to increase the sensitivity to certain BSM physics allowing for
more general interpretations of the bounds while also lowering the mass scale
which one can probe. Regarding the results presented in this paper, we would
like to encourage the experimental collaborations to use the current diboson
searches to set bounds on the anomalous couplings between the light quarks and
the $Z$ boson. It would also be very interesting to see how the degeneracy
between aTGC and $\delta V\bar{q}q$ can be resolved by considering the
production of a $Z$ in association with two jets by vector boson scattering.
In any case, we want to stress that, beyond the case of universal theories,
there exist flavour scenarios for which robust bounds on the aTGCs can only
follow from a global fit that include the effect of the $\delta V\bar{q}q$
anomalous couplings.

## Acknowledgements

We are particularly grateful to Zhengkang (Kevin) Zhang for collaborating in
the initial stages of the project. We also thank A. Azatov, S. Dawson, G.
Durieux, J. Elias-Miro, A. Falkowski, J. Gu, R.S. Gupta, G. Panico, and A.
Pomarol for helpful discussions and/or useful comments on the manuscript. C.G.
is supported by the Helmholtz Association through the recruitment initiative
program. M.R. is supported by la Caixa–Severo Ochoa grant program, by the
Spanish Ministry MEC under grants FPA2015-64041-C2-1-P, FPA2014-55613-P and
FPA2011-25948, by the Generalitat de Catalunya grant 2014-SGR-1450 and by the
Severo Ochoa excellence program of MINECO (grant SO-2012-0234). We thank the
Collaborative Research Center SFB676 of the Deutsche Forschungsgemeinschaft
(DFG), “Particles, Strings and the Early Universe”, for support.

##  Appendix A Correlations in the Higgs basis

In Figs. -888 and -887 we show the correlations between all the seven
parameters relevant for diboson production at the LHC. Since the $\chi^{2}$
function is not gaussian, these correlations are not simply related to a
covariance matrix. Instead, the 95% CL regions for each pair of parameters
with all others profiled are reported. See section 3.2 for comments.

Correlations for the current LHC data

Figure -888: One and two dimensional 95% CL constraints for the seven
parameters entering in diboson processes, using only the current LHC data in
Table 1. In yellow, all parameters are profiled. In blue, we profile over all
parameters but setting $\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. In pink, we
do an exclusive fit setting to zero all parameters that do not appear in the
plot labels.

Correlations expected at HL-LHC

Figure -887: One and two dimensional 95% CL constraints for the seven
parameters entering in diboson processes, using the $pp\to W^{+}W^{-}$ and
$pp\to WZ$ projections for 13 TeV with 3 ab-1 of integrated luminosity and
assuming a 10% systematic uncertainty. In yellow, all parameters are profiled.
In blue, we profile over all parameters but setting
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. In pink, we do an exclusive fit
setting to zero all parameters not appearing in the plot labels.

##  Appendix B Summary of LEP-1 bounds

In this appendix, we present the LEP-1 constrains obtained by profiling the
$\chi^{2}$ function obtained by Ref. [27].

#### Minimal Flavour Violation

In MVF scenarios, the vertex corrections have the following form:

$[\delta g_{L,R}^{Zu,d}]_{ij}\simeq\left(A_{L,R}^{u,d}+B_{L,R}^{u,d}\
\frac{m_{i}^{2}}{m_{3}^{2}}\right)\delta_{ij}\,,$ (38)

where $i,j=1,2,3$ stand for the family index. We are only interested in the
constraints on the light quarks $u,\,d$ that control the diboson production.
Using the results in Ref. [27] and after profiling over all other parameters
related to the electron and neutrino couplings, we arrive at

$\begin{array}[]{ccrcl}\left[\delta
g_{L}^{Zu}\right]_{11}&\,=&-0.002&\pm&0.003\\\ \left[\delta
g_{R}^{Zu}\right]_{11}&\,=&-0.003&\pm&0.005\\\ \left[\delta
g_{L}^{Zd}\right]_{11}&\,=&\,0.002&\pm&0.005\\\ \left[\delta
g_{R}^{Zd}\right]_{11}&\,=&\,0.016&\pm&0.027\end{array}\,,\quad\rho=\begin{pmatrix}1&0.43&0.52&0.23\\\
&1&0.19&0.36\\\ &&1&0.90\\\ &&&1\end{pmatrix},$ (39)

In this flavour scenario, the vertex corrections are mostly sensitive to the
$A$ coefficient in Eq. (38), while the contribution from $B$, being suppressed
by $m_{u,d}/m_{t,b}$, is negligible. The same bounds will also apply to the
$c$ and $s$ quarks since the $B$ contribution remains negligible for the
second family.

#### Flavour Universality

In FU scenarios, all the vertex corrections have the same value irrespective
of their family index, i.e.

$[\delta g_{L,R}^{Zu,d}]_{ij}=A_{L,R}^{u,d}\ \delta_{ij}\,.$ (40)

In this case the bounds for the light quarks and heavy quarks coming from
LEP-1 are the same. Using the results of Ref. [27] and after profiling over
all other parameters, the LEP-1 bounds on the vertex corrections are found to
be

$\begin{array}[]{ccrcl}\delta g_{L}^{Zu}&\,=&-0.0017&\pm&0.002\\\ \delta
g_{R}^{Zu}&\,=&-0.0023&\pm&0.005\\\ \delta
g_{L}^{Zd}&\,=&\,0.0028&\pm&0.001\\\ \delta
g_{R}^{Zd}&\,=&\,0.019&\pm&0.008\end{array}\,,\quad\rho=\begin{pmatrix}1&0.83&0.04&-0.11\\\
&1&-0.13&-0.05\\\ &&1&0.89\\\ &&&1\end{pmatrix}.$ (41)

In this case, diboson production will set bounds on all of $Z\bar{q}q$ from
just measuring the vertices for $u$ and $d$. It should be noted that, while
the bounds on the $Z\bar{u}u$ couplings are rather similar in the two MFV and
FU cases, the bounds on the $Z\bar{d}d$ couplings are about 4 times more
stringent in the FU case compared to the MFV case. This is a result of the
fact that the $b$ quark can be efficiently tagged and better discriminated
than the light quarks. On the other hand, for the case of MFV, the $Z\bar{b}b$
vertex correction gives a good constraint to the parameters $A+B$ in Eq. (38),
while $\left[\delta g_{L,R}^{Zd}\right]_{11}$ is only sensitive to $A$ and has
a much lower precision from the Z-pole observables.

#### Universal theories

For universal theories where new physics coupled to the SM degrees of freedom
via the SM currents only, the vertex corrections obey the relations 151515This
can be checked explicitly by for example writing the Higgs basis coefficients
in terms of only bosonic operators in the SILH basis, see also Ref. [30].

$\delta g_{R}^{Zu}=2(\delta g_{L}^{Zu}+\delta g_{L}^{Zd})\,,\hskip
56.9055pt\delta g_{R}^{Zd}=-(\delta g_{L}^{Zu}+\delta g_{L}^{Zd})\,,$ (42)

and only two $Zqq$ couplings are independent. We can choose them to be $\delta
g_{L}^{Zu}$ and $\delta g_{L}^{Zd}$. From the $\chi^{2}$ function
corresponding to FU theories, one can derive the bounds on these two
independent couplings

$\begin{array}[]{ccrcl}\delta g_{L}^{Zu}&\,=&-0.00010&\pm&0.00019\\\ \delta
g_{L}^{Zd}&\,=&\,0.00008&\pm&0.00018\end{array}\,,\quad\rho=\begin{pmatrix}1&-0.93\\\
&1\end{pmatrix}.$ (43)

In this scenario, the current diboson data do not set competitive bounds on
the $Z\bar{q}q$ couplings.

For completeness we show in the following the connection between $\delta
g_{L,R}^{Zu,d}$, $\delta g_{1z}$ and the oblique parameters when considering
universal theories. For $\delta g_{L,R}^{Zu,d}$ one finds:

$\delta
g_{L,R}^{Zq}\,=\,\frac{1}{2}T^{3}_{q}\,\left(\hat{T}-W-Y\,\tan^{2}\theta_{W}\right)\,+\,\frac{1}{2}Q_{q}\frac{\sin^{2}\theta_{W}}{\cos
2\theta_{W}}\left(W+\hat{T}-2\hat{S}-Y(-2+\tan^{2}\theta_{W})\right),$ (44)

which actually holds for any SM fermion. The two relations 42 are trivially
satisfied. In addition, $\delta g_{1z}$ can be written as:

$\delta g_{1z}\,=\,\frac{1}{2\cos
2\theta_{W}}\left(\hat{T}-2\,S_{\perp}+W+Y\,\tan^{2}\theta_{W}-\frac{\hat{S}-S_{\perp}}{\cos^{2}\theta_{W}}\right),$
(45)

where the oblique parameters are obtained from the coefficients of the $d=6$
operators in the SILH basis: $\hat{T}=c_{T}$, $\hat{S}=c_{W}+c_{B}$,
$W=-2c_{2W}$ and $Y=-2c_{2B}$. And $S_{\perp}$ corresponds to
$S_{\perp}=c_{W}-c_{B}$.

##  Appendix C Cross checks of the aTGC bounds

As a cross check of our methodology and our assumptions, we compared the
results of our fit with the ones presented by the experimental collaborations.
For the $pp\to WW$ channel at 8 TeV, Fig. -886 shows the comparison between
the fit of Ref. [38] by the ATLAS collaboration and the results we obtained
recasting the publicly available data. There is a good agreement. To compare
with ATLAS results, we performed a change of basis and set bounds on the
coefficients $c_{WWW},\,c_{W}$ and $c_{B}$ corresponding to the following
three operators which appear for instance in the HISZ basis, see Refs. [33,
25]:

$\displaystyle{\cal O}_{WWW}$ $\displaystyle=$
$\displaystyle\text{Tr}[W_{\mu\nu}W^{\nu\rho}W_{\rho}^{\mu}]\,,$ (46)
$\displaystyle{\cal O}_{W}$ $\displaystyle=$
$\displaystyle(D_{\mu}H)^{\dagger}W^{\mu\nu}(D_{\nu}H)\,,$ (47)
$\displaystyle{\cal O}_{B}$ $\displaystyle=$
$\displaystyle(D_{\mu}H)^{\dagger}B^{\mu\nu}(D_{\nu}H)\,.$ (48)

The Wilson coefficients of the HISZ operators entering the aTGC are related to
Higgs’ basis coefficients as follows [25]:

$\displaystyle\delta g_{1z}$ $\displaystyle=$
$\displaystyle\frac{c_{W}}{\Lambda^{2}}\,\frac{g^{2}+g^{\prime 2}}{8}v^{2}\,,$
(49) $\displaystyle\delta\kappa_{\gamma}$ $\displaystyle=$
$\displaystyle\frac{c_{W}+c_{B}}{\Lambda^{2}}\,\frac{g^{2}}{8}v^{2}\,,$ (50)
$\displaystyle\lambda_{z}$ $\displaystyle=$
$\displaystyle\frac{c_{WWW}}{\Lambda^{2}}\,\frac{3g^{4}}{8}v^{2}\,.$ (51)

In the $pp\to WZ$ channel at 8 TeV, we could not reach a similar agreement
with the ATLAS results reported in Ref. [40], but we do agree with previous
phenomenological studies [5, 6].

Figure -886: Comparison between the 95% CL contours obtained by the ATLAS
collaboration [38], and the results we obtained recasting their data.

In Fig. -885 we compare the fit on the aTGCs using the LHC diboson data
reported in Ref. [38], after profiling over the $\delta Vqq$ couplings, with
the results in Ref.[10]. In our work, the aTGC bounds are derived from a
global fit to the LHC diboson data and the $\chi^{2}$ extracted from Ref. [27]
for LEP-1. In dashed blue, we show our three parameter fit, which agrees well
with the experimental results, as shown in Fig. -886 already. Our results and
those from Ref. [10], shown in botted black, are very similar. When the
parameters $\delta V\bar{q}q$ are profiled using the LEP-1 constraints under
the FU assumption, our results (solid blue) show deviations with respect those
from Ref. [10] (solid black). The slight differences with the fit from Ref.
[10] are due to the following: i) only the last bin of the experimental
distribution is used in Ref. [10] while we use all of them, ii) the procedure
itself to set the bounds for the aTGC in Ref. [10] is different which could
also create some discrepancy with our results. To asses the first point, we
show in red our fit after profiling over the quark couplings when only the
last bin is used. We find that this has a better agreement with Ref. [10],
nonetheless not taking into account the subleading bins spoils our agreement
with the ATLAS result.

Figure -885: 95% CL contours for the aTGCs marginalizing over all other
parameters. A three parameter fit (dashed or dotted) with $\delta V\bar{q}q=0$
and a seven parameter fit (solid lines) combined LHC diboson data and LEP-1
data are performed. For the later fit, a FU setup is considered. The blue
lines are our results, in black the results obtained in Ref. [10], and in red,
our results taking into account only the last bin.

##  Appendix D Comparison of HEP parameter bounds

To compare with previous works studying diboson production, in Fig. -884 we
present the HL-LHC bounds for the high energy parameters (HEP) defined in Ref.
[12]. These HEP appear in the helicity amplitudes of Eqs. (6), (7). In order
to rewrite the Higgs basis in terms of the HEP, we perform a change of basis
to in $\chi^{2}$ function inverting the following relations:

$\displaystyle a_{q}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{m_{W}^{2}}\big{[}\delta g_{L}^{Zu}-\delta
g_{L}^{Zd}-c_{W}^{2}\,\delta g_{1z}\big{]},\,$ (52) $\displaystyle
a_{q}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{3\,m_{W}^{2}}\big{[}3\,(\delta g_{L}^{Zu}+\delta
g_{L}^{Zd})+(\delta\kappa_{\gamma}\,t_{W}^{2}-\delta
g_{1z}\,s_{W}^{2})\,\big{]},\,$ (53) $\displaystyle a_{u}$ $\displaystyle=$
$\displaystyle-\frac{4\,g^{2}}{3\,m_{W}^{2}}\big{[}\frac{3}{2}\,\delta
g_{R}^{Zu}+(\delta\kappa_{\gamma}\,t_{W}^{2}-\delta
g_{1z}\,s_{W}^{2})\big{]},\,$ (54) $\displaystyle a_{d}$ $\displaystyle=$
$\displaystyle\frac{2\,g^{2}}{3\,m_{W}^{2}}\big{[}-3\,\delta
g_{R}^{Zd}+(\delta\kappa_{\gamma}\,t_{W}^{2}-\delta
g_{1z}\,s_{W}^{2})\big{]}.$ (55)

The $\chi^{2}$ function then becomes a function of the four HEP, and
$\lambda_{\gamma}$, and two other orthogonal combinations which we call
$b_{1},\,b_{2}$. These orthogonal combinations appear in the subleading
amplitudes shown in Eqs. (9), (LABEL:eq:subWZ). Figure -884 shows in red and
blue the derived $\chi^{2}$ function for $a_{q}^{(3)}$ assuming
$\delta_{syst}=5\%$. 161616See section 4 for the definition of
$\delta_{syst}$. The blue and red colours correspond to the case where all the
bins of the differential distributions are used (blue) and the one where only
the last bin is used (red). Clearly, the actual bound is not entirely
dominated by the most energetic bin and all the bins do contribute to setting
the bound. We explicitly studied three different cases: i) in dashed, we set
$\lambda_{\gamma}$, the three remaining HEP and the orthogonal directions
$b_{1},\,b_{2}$ to zero, ii) in solid we set $b_{1},\,b_{2}$ and
$\lambda_{\gamma}$ to zero but profile over the three remaining HEP, iii) in
dotted we profile over all the parameters. We find that, as expected, the four
HEP parameters are not very correlated among them, and therefore the solid and
dashed lines differ by a small amount. On the other hand, including or not the
subleading terms $b_{1},\,b_{2}$, which appear in the amplitudes shown in Eqs.
(9), (LABEL:eq:subWZ), makes a significant change. In general, we expect the
subleading terms to be relatively important when the quadratic dimension-six
amplitudes dominate the interference with the SM. On the other hand if the
interference with the SM dominates, we expect these pieces to have an extra
suppression coming from the SM amplitudes. In vertical orange and green lines
we present the HL-LHC prospects of the leptonic $pp\to WZ$ and semi-leptonic
$pp\to WV$ obtained in Refs. [12, 23]; we differentiate in their case with
solid and dashed lines two different assumptions on the systematic errors. If
the new observables proposed by Refs. [12, 23] are implemented, they will be
able to set stronger bounds on $a_{q}^{(3)}$ by at least a factor 2.

Figure -884: $\chi^{2}$ for $a_{q}^{(3)}$ ffrom the HL-LHC projections and
comparison with the 95% bounds obtained in Refs. [12, 23].

The 95% CL bounds for the HEP that we get when marginalizing over all the
other parameters, i.e. three HEP, $b_{1},b_{2}$ and $\lambda_{\gamma}$ are the
following:

$\displaystyle\Delta a_{q}^{(3)}$ $\displaystyle=$
${}^{+0.11}_{-0.18}\,,\hskip 56.9055pt\Delta a_{q}^{(1)}\
=\,^{+0.34}_{-0.32}\,,$ (56) $\displaystyle\Delta a_{u}$ $\displaystyle=$
${}^{+0.36}_{-0.47}\,,\hskip 56.9055pt\Delta a_{d}\ =\ ^{+0.60}_{-0.55}\,.$
(57)

On the other hand, Ref. [12] gets $\Delta a_{q}^{(3)}\simeq\pm 0.04\,(0.15)$
for systematics of 5% (100%) and Ref. [23] finds $\Delta a_{q}^{(3)}\simeq\pm
0.03\,(0.04)$ for systematics of 3% (10%).

Another way to compare with previous works is to set bounds on the
$a_{q}^{(3)},\,a_{q}^{(1)}$ plane but considering only universal theories
where the oblique parameters $W,\,Y$ are negligible, see Refs. [12, 16]. In
the SILH basis using the conventions in Ref. [25], one obtains

$\displaystyle a_{q}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{m_{W}^{2}}\,(c_{HW}+c_{W}-2c_{2W})\,,\hskip
28.45274pta_{u}=\frac{3\,g^{2}}{m_{W}^{2}}c_{Hu}+\frac{4\,g^{\prime
2}}{3\,m_{W}^{2}}(c_{B}+c_{HB}+2c_{2B})\,,$ (58) $\displaystyle a_{q}^{(1)}$
$\displaystyle=$ $\displaystyle-\frac{g^{\prime
2}}{3m_{W}^{2}}\,(c_{B}+c_{HB}+2c_{2B})\,,\hskip
20.77039pta_{d}=\frac{g^{2}}{m_{W}^{2}}c_{Hd}-\frac{2\,g^{\prime
2}}{3\,m_{W}^{2}}(c_{B}+c_{HB}+2c_{2B})\,,$

therefore, for universal theories and neglecting $W=-2c_{2W}$ and $Y=-2c_{2B}$
the HEP are:

$\displaystyle a_{q}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{m_{W}^{2}}\,(c_{HW}+c_{W})\,,\hskip
28.45274pta_{q}^{(1)}=-\frac{1}{4}a_{u}=\frac{1}{2}a_{d}=-\frac{g^{\prime
2}}{3m_{W}^{2}}\,(c_{B}+c_{HB})\,,$ (59)

which in terms of $\hat{S}$, $\delta g_{1z}$ and $\delta\kappa_{\gamma}$,
using the conventions in Ref. [25], can be written as:

$\displaystyle a_{q}^{(3)}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{m_{W}^{2}}(\delta
g_{1z}\,c_{W}^{2}+\frac{g^{\prime 2}}{g^{2}-g^{\prime 2}}\hat{S}),\hskip
28.45274pta_{q}^{(1)}=-a_{q}^{(3)}\,\frac{t_{W}^{2}}{3}-\frac{g^{\prime
2}}{3m_{W}^{2}}(\delta\kappa_{\gamma}-\hat{S})\,.$ (60)

In Refs. [12, 16] for convenience they choose as independent directions
$\delta g_{1z}\,c_{W}^{2}+\frac{g^{\prime 2}}{g^{\prime 2}-g^{2}}\hat{S}$ and
$\delta\kappa_{\gamma}-\hat{S}$. We show our bounds in this plane in Fig.
-883.

Figure -883: 95%CL constraints on universal theories with $W,Y\ll 1$. Left:
Comparison of the LEP constraints with the ones extracted from the current LHC
diboson data (dotted) and the HL-LHC projections (solid). Right: Constraints
of this work compared with the ones in Refs. [12, 16]. This fit for universal
theories agrees with the 3-parameter fit of Fig. -893 when only the aTGC
couplings are considered and the $\delta V\bar{q}q$ deviations are set to
zero.

##  Appendix E High energy amplitudes in the Warsaw basis

Using the dictionary of Ref. [25] we express the amplitude shown in Eqs.
(6)–(7) in the Warsaw basis. The high energy amplitudes for $pp\to WW$ are
given by:

$\displaystyle\mathcal{M}(u_{L}u_{L}\rightarrow LL;00)$ $\displaystyle=$
$\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}}{4\,s_{W}^{2}}\sin\theta\,\left[\frac{v^{2}}{\Lambda^{2}}(\omega_{\phi
q}^{(1)}+\omega_{\phi q}^{(3)})\right]\,,$ (61)
$\displaystyle\mathcal{M}(d_{L}d_{L}\rightarrow LL;00)$ $\displaystyle=$
$\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}}{4\,s_{W}^{2}}\sin\theta\,\left[\frac{v^{2}}{\Lambda^{2}}(\omega_{\phi
q}^{(1)}-\omega_{\phi q}^{(3)})\right]\,,$
$\displaystyle\mathcal{M}(u_{R}u_{R}\rightarrow RR;00)$ $\displaystyle=$
$\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}}{4\,s_{W}^{2}}\sin\theta\,\left[-\frac{v^{2}}{\Lambda^{2}}\omega_{\phi
u}\right]\,,$ $\displaystyle\mathcal{M}(d_{R}d_{R}\rightarrow RR;00)$
$\displaystyle=$ $\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}}{4\,s_{W}^{2}}\sin\theta\,\left[-\frac{v^{2}}{\Lambda^{2}}\omega_{\phi
d}\right]\,,$ $\displaystyle\mathcal{M}(LL;\pm\pm)$ $\displaystyle=$
$\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{3e^{3}}{4\,t_{W}s_{W}}\,T_{q}^{3}\,\sin\theta\,\left[-\frac{v^{2}}{\Lambda^{2}}\,\omega_{W}\right]\,,$

while for $pp\to WZ$, these are:

$\displaystyle\mathcal{M}(LL;00)$ $\displaystyle=$ $\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{e^{2}}{2\sqrt{2}s_{W}^{2}c_{W}}\sin\theta\left[-\frac{v^{2}}{\Lambda^{2}}\omega_{\phi
q}^{(3)}\right]\,,$ $\displaystyle\mathcal{M}(LL;\pm\pm)$ $\displaystyle=$
$\displaystyle
i\frac{\hat{s}}{m_{W}^{2}}\frac{3e^{3}}{4\sqrt{2}s_{W}^{2}}\,\sin\theta\left[\frac{v^{2}}{\Lambda^{2}}\omega_{W}\right].$
(62)

##  Appendix F Summary tables

In this appendix, we report the results of the various fits performed in this
paper.

### Constraints on $\delta V\bar{q}q$

We first report the bounds on $\delta V\bar{q}q$ under various assumptions on
the aTGCs. See Tables 2, 3, 4, 5.

1$\sigma$ bounds on $\delta V\bar{q}q$ from current LHC data (three aTGCs
profiled)
---
$\times 10^{3}$ | Diboson | LEP (MFV) | LEP (FU) | Comb. (MFV) | Comb. (FU)
$[\delta g_{L}^{Zu}]_{11}$ | $-7.9\pm 10$ | $-1.9\pm 3.1$ | $-1.7\pm 2.1$ | $-1.9\pm 2.8$ | $-0.9\pm 2.0$
$[\delta g_{R}^{Zu}]_{11}$ | $-40\pm 24$ | $-2.6\pm 5$ | $-2.3\pm 4.6$ | $-1.3\pm 4.4$ | $-0.9\pm 4.3$
$[\delta g_{L}^{Zd}]_{11}$ | $-18\pm 7$ | $2\pm 5.4$ | $2.8\pm 1.5$ | $-2\pm 2.5$ | $0.8\pm 1.2$
$[\delta g_{R}^{Zd}]_{11}$ | $20.3\pm 14.2$ | $16\pm 27$ | $20\pm 7.7$ | $-1.8\pm 7.9$ | $8.6\pm 5.6$
Table 2: Constraints $(\times 10^{3})$ on the $\delta V\bar{q}q$ vertex
corrections from a seven parameter global fit combining LHC diboson data and
LEP-1 measurements. The first column gives the bounds using the LHC diboson
data alone. The second and third columns report the LEP-1 bounds derived in
Ref. [27] under the MFV and FU assumptions respectively. Finally, the last two
columns show the combination of the current LHC and LEP-1 data for the two
flavour assumptions. 1$\sigma$ bounds on $\delta V\bar{q}q$ from current LHC
data ($\delta\kappa_{\gamma}=\lambda_{\gamma}=0$)
---
$\times 10^{3}$ | Diboson | LEP (MFV) | LEP (FU) | Comb. (MFV) | Comb. (FU)
$[\delta g_{L}^{Zu}]_{11}$ | $6.4\pm 6.4$ | $-1.9\pm 3.1$ | $-1.7\pm 2.1$ | $-1.5\pm 2.7$ | $-0.3\pm 1.8$
$[\delta g_{R}^{Zu}]_{11}$ | $4.3\pm 6.4$ | $-2.6\pm 5$ | $-2.3\pm 4.6$ | $-0.3\pm 3.9$ | $0.6\pm 3.8$
$[\delta g_{L}^{Zd}]_{11}$ | $-8.7\pm 5.2$ | $2\pm 5.4$ | $2.8\pm 1.5$ | $-2\pm 2.5$ | $0.6\pm 1.1$
$[\delta g_{R}^{Zd}]_{11}$ | $-2.1\pm 8$ | $16\pm 27$ | $20\pm 7.7$ | $-2.7\pm 7.7$ | $7.7\pm 5.4$
Table 3: Constraints $(\times 10^{3})$ on the $\delta V\bar{q}q$ vertex
corrections from a five parameter global fit combining LHC diboson data and
LEP-1 measurements, setting $\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. The
first column gives the bounds using the LHC diboson data from Table 1 setting
$\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. The first column gives the bounds
using the LHC diboson data alone. The second and third columns report the
LEP-1 bounds derived in Ref. [27] under the MFV and FU assumptions
respectively. Finally, the last two columns show the combination of the
current LHC and LEP-1 data for the two flavour assumptions. 1$\sigma$ bounds
on $\delta V\bar{q}q$ from current LHC data ($\delta
g_{1z}=\delta\kappa_{\gamma}=\lambda_{\gamma}=0$)
---
$\times 10^{3}$ | Diboson | LEP (MFV) | LEP (FU) | Comb. (MFV) | Comb. (FU)
$[\delta g_{L}^{Zu}]_{11}$ | $0.6\pm 5.0$ | $-1.9\pm 3.1$ | $-1.7\pm 2.1$ | $-0.7\pm 2.6$ | $1.1\pm 1.6$
$[\delta g_{R}^{Zu}]_{11}$ | $1.5\pm 6.1$ | $-2.6\pm 5$ | $-2.3\pm 4.6$ | $1.5\pm 3.6$ | $3.5\pm 3.4$
$[\delta g_{L}^{Zd}]_{11}$ | $-5.3\pm 4.6$ | $2\pm 5.4$ | $2.8\pm 1.5$ | $-2.8\pm 2.4$ | $0.1\pm 1.1$
$[\delta g_{R}^{Zd}]_{11}$ | $-0.71\pm 8$ | $16\pm 27$ | $20\pm 7.7$ | $-6.2\pm 6.8$ | $5.0\pm 5.4$
Table 4: Constraints $(\times 10^{3})$ on the $\delta V\bar{q}q$ vertex
corrections from a five parameter global fit combining LHC diboson data and
LEP-1 measurements, setting $\delta
g_{1z}=\delta\kappa_{\gamma}=\lambda_{\gamma}=0$. The first column gives the
bounds using the LHC diboson data alone. The second and third columns report
the LEP-1 bounds derived in Ref. [27] under the MFV and FU assumptions
respectively. Finally, the last two columns show the combination of the
current LHC and LEP-1 data for the two flavour assumptions. 1$\sigma$ bounds
on $\delta V\bar{q}q$ expected at HL-LHC
---
$\times 10^{3}$ | aTGCs profiled | no loop $(\delta\kappa_{\gamma}=\lambda_{\gamma}=0)$ | no aTGCs | exclusive fit
$[\delta g_{L}^{Zu}]_{11}$ | $\pm 2.5$ | $\pm 1.5$ | $\pm 1.4$ | $\pm 0.5$
$[\delta g_{R}^{Zu}]_{11}$ | $\pm 4.0$ | $\pm 1.7$ | $\pm 1.5$ | $\pm 1.1$
$[\delta g_{L}^{Zd}]_{11}$ | $\pm 1.5$ | $\pm 1.2$ | $\pm 1.2$ | $\pm 0.45$
$[\delta g_{R}^{Zd}]_{11}$ | $\pm 4.0$ | $\pm 2.5$ | $\pm 2.5$ | $\pm 1.8$
Table 5: Expected constraints ($\times 10^{3}$) at HL-LHC on the $\delta
V\bar{q}q$ vertex corrections. The constraints are obtained from the
projections at HL-LHC for the $pp\to W^{+}W^{-}\to\ell\nu\ell\nu$ channel
combined with the LEP-1 constraints for a MFV setup. The first column gives
the constraints resulting from a seven parameter fit. In the second, the two
aTGCs usually generated at the loop level are set to zero. In the third
column, all the three aTGCs are set to zero. Finally, the last column reports
the constraints obtained from an exclusive fit with only one parameter
considered at a time.

### Constraints on the aTGCs

We now report the bounds on the aTGCs under various assumptions on $\delta
V\bar{q}q$. These are shown in Tables 6 and 7.

1$\sigma$ bounds on aTGC from current LHC data
---
$\times 10^{3}$ | $\delta g_{L,R}^{Zu,d}=0$ |  $\delta g_{L,R}^{Zu,d}=$MFV |  $\delta g_{L,R}^{Zu,d}=$FU
$\delta\kappa_{\gamma}$ | $12\pm 31$ | $18\pm 35$ | $24\pm 35$
$\delta g_{1z}$ | $-7\pm 4$ | $-7\pm 5$ | $-9\pm 5$
$\lambda_{\gamma}$ | $0\pm 6$ | $0\pm 6$ | $0\pm 6$
Table 6: Constraints $(\times 10^{3})$ on the anomalous triple gauge couplings
from the current LHC diboson data. The first column corresponds to the
traditional diboson analysis that considers only aTGCs and sets to zero all
anomalous fermion-gauge vertices $\delta V\bar{q}q=0$. The next two columns
show the effect of letting these anomalous fermion-gauge vertices float,
assuming either a MFV or a FU setup respectively. 1$\sigma$ bounds on aTGC
expected at HL-LHC
---
$\times 10^{3}$ | $\delta g_{L,R}^{Zu,d}=0$ |  $\delta g_{L,R}^{Zu,d}=$MFV |  $\delta g_{L,R}^{Zu,d}=$FU
$\delta\kappa_{\gamma}$ | $\pm 10$ | $\pm 22$ | $\pm 20$
$\delta g_{1z}$ | $\pm 1.5$ | $\pm 3.5$ | $\pm 3.0$
$\lambda_{\gamma}$ | $\pm 2.2$ | $\pm 2.3$ | $\pm 2.2$
Table 7: Constraints ($\times 10^{3}$) on the anomalous triple gauge couplings
using the projections at HL-LHC of the $pp\to W^{+}W^{-}\to\ell\nu\ell\nu$
channel. The first column corresponds to the traditional diboson analysis that
considers only aTGCs and sets to zero all anomalous fermion-gauge vertices
$\delta V\bar{q}q=0$. The next two columns show the effect of letting these
anomalou fermion-gauge vertices float, assuming either a MFV or a FU setup
respectively.

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