# Composite effective field theory signal in case of searching for neutral
triple gauge couplings with $\boldsymbol{{ZZ\rightarrow\ell\ell\nu\nu}}$
production

Artur E. Semushina,b,*, Evgeny Yu. Soldatova,†

###### Abstract

In this work manifestations of physics beyond the Standard Model are
parameterized with higher-dimension operators and Wilson coefficients using
effective field theory. In order to set more stringent experimental limits on
the Wilson coefficients it is crucial to use new methods of sensitivity
increasing that work independently with luminosity growth. Method of composite
anomalous signal allows one to set the limits on Wilson coefficients more
precisely and stringent by accounting for impact of EFT operators on
background processes in addition to the conventional anomalous contribution
from the signal process. This work presents aforementioned methodology applied
to the ${ZZ\rightarrow\ell\ell\nu\nu}$ production, that is sensitive to the
neutral triple couplings of gauge bosons $Z$ and $\gamma$. It is found that
the main background BSM contribution comes from
${WZ\rightarrow\ell\nu\ell\ell}$ background. Improvement of the limits on the
Wilson coefficients due to this background depends on the coefficient and is
up to 57.5% (59.1%) for one- (two-) dimensional limits in linear + quadratic
effective field theory model and up to 94.4% for one-dimensional limits in
linear effective field theory model.

##  1 Introduction

The Standard Model (SM) of particle physics combines knowledge about all known
elementary particles and their interactions. The first observation of the
Higgs boson has been made in 2012 at the Large Hadron Collider (LHC) [1, 2]
and become a very strong proof of the correctness of the SM. Furthermore, the
SM was precisely tested in high-energy-physics experiments by studying rare
phenomena. These studies yielded in a good agreement between theoretical
predictions and experimental data [3].

However, the SM has internal inconsistencies, such as dependence on a set of
free parameters and fine-tuning problem. Moreover, the SM do not describe
gravity and some observed phenomena. For instance, it fails to describe
neutrino oscillations and baryon asymmetry of the Universe. Due to the
aforementioned facts, the SM should be considered as an effective theory, that
is low-energy approximation of the more general theory. Therefore,
experimental tests of the SM are usually accompanied by searches for beyond-
the-SM (BSM) phenomena, so-called new physics, and its observation is one of
the main goals of the modern high-energy-physics experiments.

New physics can be searched for in two ways. Direct approach implies search
for new particles, whereas indirect approach is based on looking for
deviations from the SM in the interactions of already known particles. Since
the evidences of new particles presence at the currently available energies
was not found [4, 5], the indirect way becomes more prospective. The searches
for anomalous couplings refer to this way and allow one to look for
manifestations of heavy new physics at currently accessible energies. A
convenient way to parameterize the anomalous couplings is an effective field
theory (EFT) [6, 7], that uses higher-dimension operators. Free parameters of
this approach, Wilson coefficients, can be constrained experimentally. Non-
zero Wilson coefficients affect all physical processes, and experimental data
is a composite of contributions from signal and background processes.
Therefore, it is important to study the impact of EFT on background processes
in addition to the conventional impact of EFT on signal process, since
background EFT contributions can significantly change the experimental
sensitivity [8].

EFT impact on background processes strongly depends on the channel being
studied and the choice of EFT operator basis. In this work, EFT contributions
from backgrounds are studied on the example of ${ZZ\rightarrow\ell\ell\nu\nu}$
production at the LHC. Studies of this process are usually interpreted in
terms of anomalous neutral triple gauge couplings (nTGCs) due to the large
sensitivity [9, 10]. Therefore, dimension-eight operators affecting nTGCs are
used in this work.

##  2 Effective field theory framework

EFT is a powerful tool to extend the SM in a model-independent way. It
parameterizes manifestations of new physics with heavy energy scale $\Lambda$
at currently accessible energies as anomalous couplings. For this purpose, EFT
uses operators of dimension greater than four, which is the dimension of the
SM Lagrangian $\mathcal{L}_{\text{SM}}$. The resulting EFT Lagrangian can be
written as

$\mathcal{L}=\mathcal{L}_{\text{SM}}+\sum\limits_{d>4}\sum\limits_{i}\frac{C_{i}^{(d)}}{\Lambda^{d-4}}\mathcal{O}_{i}^{(d)},$
(1)

where $\mathcal{O}_{i}^{(d)}$ is the $i$-th dimension-$d$ operator, and
$C_{i}^{(d)}/\Lambda^{d-4}$ is the corresponding Wilson coefficient. The
operators are constructed out of the SM fields and required to respect the SM
gauge symmetries. Dimension-odd operators necessarily contain fermions and,
therefore, are not valid for studying anomalous bosonic couplings. In the SM
all the Wilson coefficients are zero, whereas their non-zero values imply the
presence of new physics. This new physics may arise from new heavy particles
interacting with the SM particles and, therefore, effectively changing the
couplings between the SM particles with energies lower than the new physics
scale. Integrating the new heavy fields out from the Lagrangian of new physics
model, one can obtain a model-dependent low-energy effective Lagrangian (for
example, Euler–Heisenberg Lagrangian [11]). Wilson coefficients can be
reinterpreted in terms of SM parameters and new particles’ masses by matching
model-independent and model-dependent Lagrangians [12, 13]. Therefore, limits
on Wilson coefficients can be transformed into the limits on new particles
parameters. This can constrain some new physics models, and it is crucial to
set experimental limits on Wilson coefficients correctly and to make them more
stringent if it is possible.

Neutral triple gauge couplings, i.e. $ZZZ$, $ZZ\gamma$, $Z\gamma\gamma$ and
$\gamma\gamma\gamma$, are zero in the SM. They can be added to the theory
Lagrangian using EFT. Dimension-six operators do not predict such couplings,
therefore, dimension-eight operators are used to study nTGCs. For the study,
carried out in this work, the following six dimension-eight operators are
considered:

$\displaystyle\mathcal{O}_{G\pm}=g^{-1}\tilde{B}_{\mu\nu}W^{a\mu\rho}\left(D_{\rho}D_{\lambda}W^{a\nu\lambda}\pm
D^{\nu}D^{\lambda}W^{a}_{\lambda\rho}\right),$ (2)
$\displaystyle\mathcal{O}_{\tilde{B}W}=i\Phi^{\dagger}\tilde{B}_{\mu\nu}\hat{W}^{\mu\rho}\\{D_{\rho},D^{\nu}\\}\Phi+\text{h.c.},$
(3)
$\displaystyle\mathcal{O}_{BW}=i\Phi^{\dagger}B_{\mu\nu}\hat{W}^{\mu\rho}\\{D_{\rho},D^{\nu}\\}\Phi+\text{h.c.},$
(4)
$\displaystyle\mathcal{O}_{BB}=i\Phi^{\dagger}B_{\mu\nu}B^{\mu\rho}\\{D_{\rho},D^{\nu}\\}\Phi+\text{h.c.},$
(5)
$\displaystyle\mathcal{O}_{WW}=i\Phi^{\dagger}\hat{W}_{\mu\nu}\hat{W}^{\mu\rho}\\{D_{\rho},D^{\nu}\\}\Phi+\text{h.c.}$
(6)

This basis consists of two pure gauge operators [14, 15] and four mixed Higgs-
gauge operators [16]. The former operators are classified under the assumption
of $CP$-conservation, whereas the latter ones form a full basis only under the
assumption that final-state $Z$ is on-shell. All these operators violate
$C$-symmetry. $CP$-symmetry is conserved by $\mathcal{O}_{G+}$,
$\mathcal{O}_{G-}$, $\mathcal{O}_{\tilde{B}W}$ and violated by
$\mathcal{O}_{BW}$, $\mathcal{O}_{BB}$, $\mathcal{O}_{WW}$ operators. Each
operator affects a set of vertices, changing corresponding vertex functions
compared to the SM ones. Effect of considered six operators on different
triple gauge-boson couplings is summarized in Table 1.

Table 1: Effect of dimension-eight EFT operators on triple gauge couplings; affected couplings are marked with $\circ$. Operator |  $ZZZ$, $ZZ\gamma$, $Z\gamma\gamma$ | $\gamma\gamma\gamma$ | $WWZ$ | $WW\gamma$
---|---|---|---|---
$\mathcal{O}_{G\pm}$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
$\mathcal{O}_{\tilde{B}W}$ | $\circ$ |  | $\circ$ | $\circ$
$\mathcal{O}_{BW}$ | $\circ$ |  | $\circ$ | $\circ$
$\mathcal{O}_{WW}$ | $\circ$ |  | $\circ$ |
$\mathcal{O}_{BB}$ | $\circ$ |  |  |

Usually the Lagrangian is parameterized by one or two operators at a time,
that yields in one- or two-dimensional constraints on the Wilson coefficients.
In case if the Lagrangian is parameterized by one operator and if the process
contains only one anomalous vertex, process squared amplitude and, therefore,
cross section contains three terms:

$|\mathcal{A}|^{2}=|\mathcal{A}_{\text{SM}}+(C/\Lambda^{4})\mathcal{A}_{\text{BSM}}|^{2}=|\mathcal{A}_{\text{SM}}|^{2}+(C/\Lambda^{4})2\text{Re}\mathcal{A}_{\text{SM}}^{\dagger}\mathcal{A}_{\text{BSM}}+(C/\Lambda^{4})^{2}|\mathcal{A}_{\text{BSM}}|^{2}.$
(7)

The first term comes purely from the SM and does not depend on the Wilson
coefficient. The second term is interference between SM and new physics
contribution, so-called interference (linear) term. The third term is the
quadratic term, that comes purely from new physics. In the case of two-
dimensional parameterization, the process squared amplitude contains six
terms. In addition to the terms that are analogous to the ones from one-
dimensional parameterization, there is a cross-term, representing interference
between two operators and proportional to the product of two Wilson
coefficients. This term can significantly change the sensitivity.

Since energy scale $\Lambda$ is heavy, the quadratic and cross terms
($\propto\Lambda^{-8}$) should be suppressed compared to the interference
terms ($\propto\Lambda^{-4}$). Moreover, interference between dimension-twelve
operators and SM is also $\propto\Lambda^{-8}$, that can lead to the change of
the sensitivity due to the statistical correlation between dimension-eight and
dimension-twelve Wilson coefficient. However, for $CP$-odd operators
interference is zero without accounting for special $CP$-sensitive variables,
and for $CP$-even operators effect of interference suppression due to the
allowed boson polarizations is presented [16]. Therefore, in this work a model
with all terms from Eq. (7) (linear + quadratic model) is baseline and used
under the assumption that interference contributions from dimension-twelve
operators are suppressed compared to the quadratic contributions from
dimension-eight operators.

Additionally, it is needed to show effect of background processes on linear
term, since at the current experimental sensitivity quadratic term dominates.
For this purpose, a model with dropped quadratic term (linear model) is used
for 1D limits on Wilson coefficients on $CP$-even operators only.

##  3 Modelling of the physical processes

EFT impact on background processes is studied for
${ZZ\rightarrow\ell\ell\nu\nu}$ production in $pp$ collisions at $\sqrt{s}=13$
TeV. Integrated luminosity of 140 fb-1, corresponding to the data collected by
the ATLAS experiment during LHC Run II [17], is used for the calculations.
Examples of the Feynman diagrams of this process production in the SM and via
nTGC are shown in Fig. 1. The main background for this process is
${WZ\rightarrow\ell\nu\ell\ell}$ production, where one lepton is either not
identified or a $\tau$, decaying to jets or non-identified charged lepton and
neutrinos. Sizeable background comes from non-resonant $\ell\ell$ production,
and in this work non-resonant background is a combination of
${WW\rightarrow\ell\nu\ell\nu}$, ${Z\rightarrow\tau\tau}$, $tt$ and $tW$
processes production. Possible mismeasurement of jet energies leads to the
fake missing transverse energy, that emulates neutrinos. This background
dominantly comes from ${Z(\rightarrow\ell\ell)\text{ + jets}}$ production.
Additional background is ${ZZ\rightarrow 4\ell}$ production, where only two
charged leptons are identified by the detector. Other backgrounds are not
accounted, since they yield small contribution [9], whereas this work uses
only toy model in order to study EFT impact on background processes.

Figure 1: Examples of the Feynman diagrams for ${ZZ\rightarrow\ell\ell\nu\nu}$
production via the SM vertices (left) and via a neutral triple gauge coupling
(right).

All aforementioned backgrounds are affected by non-zero Wilson coefficients of
operators from Eqs. (2)–(6) at the tree level. However, some backgrounds
require one ($tW$, ${Z(\rightarrow\ell\ell)\text{ + jets}}$) or two
(${Z\rightarrow\tau\tau}$, $tt$) additional jets for being produced via triple
gauge coupling. BSM contributions from such backgrounds should be small, but
they are accounted in this study. The exception is impact of
$\mathcal{O}_{BB}$ operator on processes ${WZ\rightarrow\ell\nu\ell\ell}$,
${WW\rightarrow\ell\nu\ell\nu}$ and $tW$, that is zero at the leading order,
as it can be seen in Table 1. $\mathcal{O}_{BB}$ affects these processes under
the requirement of two additional jets, however in this case anomalous quartic
gauge couplings and Feynman diagrams with two anomalous vertices are
presented. Thus, EFT contributions from ${WZ\rightarrow\ell\nu\ell\ell}$,
${WW\rightarrow\ell\nu\ell\nu}$ and $tW$ productions for $\mathcal{O}_{BB}$
operator are assumed to be negligible.

All the processes were modelled using Monte Carlo event generator
MadGraph5_aMC@NLO [18, 19]. In order to model the BSM contributions, a
decomposition technique in MadGraph5_aMC@NLO was used, that allows generating
the events, corresponding to the different squared amplitude terms, separately
[20]. Parton distribution functions, calculated by the NNPDF collaboration
[21], were used in the computations. Parton shower, hadronization and
underlying event was simulated using Pythia8 [22]. For some SM processes
accounting the events with additional jets significantly changes the cross
section. So, Pythia8 was also used to merge matrix element and parton shower
for such processes with CKKW-L scheme [23]. Finally, interaction of particles
with a typical LHC detector were simulated using Delphes3 [24]. Note that in
Monte Carlo simulation $\ell$ for ${Z(\rightarrow\ell\ell)\text{ + jets}}$
background means $e$ or $\mu$, whereas for other processes $\ell$ includes
also $\tau$.

Object reconstruction and event selection criteria in this work are based on
the study of ${ZZ\rightarrow\ell\ell\nu\nu}$ production with 36.1 fb-1 of
$pp$-collisions at $\sqrt{s}=13$ TeV by the ATLAS collaboration [9]. Some
criteria are changed in order to simplify the toy model and since this study
does not use full simulation of the detector. The criteria of the event
selection are summarized in Table 2 111This work uses right-handed coordinate
system with origin at the centre of the detector, $x$\- and $y$-axes directed
to the centre of the LHC and upwards, and $z$-axis located along the beam
pipe. Transverse momentum $\vec{p}_{\text{T}}$ is calculated in the $xy$
plane. Variable $\Delta R=\sqrt{(\Delta\eta)^{2}+(\Delta\varphi)^{2}}$
measures angular distance between two objects, where
$\eta=-\log\tan(\theta/2)$ is the pseudorapidity.. After applying these
criteria, normalization of the SM ${Z(\rightarrow\ell\ell)\text{ + jets}}$
background was found too large. It can be explained by the fact that this
background is modelled badly and usually estimated using data-driven
techniques [25], and detector simulation in Delphes3 reproduces this
background incorrectly. So, for the SM ${Z(\rightarrow\ell\ell)\text{ +
jets}}$ background the shape was taken from Monte Carlo modelling, and the
normalization was taken from the ATLAS study [9]. Moreover, for non-resonant
backgrounds both shape and normalization were modelled badly due to the low
statistics of Monte Carlo samples. Therefore, for this background shape was
taken the same as for ${ZZ\rightarrow\ell\ell\nu\nu}$ production, and
normalization was also taken from the ATLAS study [9].

Table 2: Event selection criteria. Two leptons of same flavor and opposite
signs
---
$p_{\text{T}}^{\ell_{1}}>30$ GeV, $p_{\text{T}}^{\ell_{2}}>20$ GeV
$p_{\text{T}}^{\ell\ell}>150$ GeV
$76<m_{\ell\ell}<106$
$\Delta R_{\ell\ell}>1.9$
$E_{\text{T}}^{\text{miss}}>120$ GeV
$N_{j}\geq 0$, $N_{b\text{-jet}}=0$
$\left|\sum\limits_{\text{objects}}\mathbf{p}_{\text{T}}^{\leavevmode\nobreak\
j}\right|/\sum\limits_{\text{objects}}p_{\text{T}}^{j}>0.65$

For setting limits on the Wilson coefficients, momentum of two leptons
$p_{\text{T}}^{\ell\ell}$ is used as a sensitive variable due to the large
sensitivity of this variable to anomalous couplings, especially in
high-$p_{\text{T}}^{\ell\ell}$ region [9, 10]. Distributions by this variable
for three different Wilson coefficients with illustrations of contributions
from each SM background are presented in Fig. 2. Fig. 3 shows contributions to
the quadratic term from signal and each background processes. Fig. 4 shows
contributions to the interference term from signal and main
(${WZ\rightarrow\ell\nu\ell\ell}$) background processes. Due to the large
Monte Carlo uncertainties, binning of Fig. 4 was changed and backgrounds
except ${WZ\rightarrow\ell\nu\ell\ell}$ were dropped. Conventional method for
setting the limits on the Wilson coefficients uses BSM contributions from the
signal process only, whereas in general case EFT signal is composed of
contributions from different processes.

Figure 2: $p_{\text{T}}^{\ell\ell}$ distributions of expected event yields.
Filled histogram shows contributions from SM signal and each background
processes. Orange (red) line represents expected event yields if one Wilson
coefficient is non-zero for the case if BSM contributions from signal process
only (signal and all background processes) are taken into account:
$C_{G+}/\Lambda^{4}$ (left), $C_{G-}/\Lambda^{4}$ (center) and
$C_{\tilde{B}W}/\Lambda^{4}$ (right). At the lower panel ratio of red and
orange lines is presented.

Figure 3: $p_{\text{T}}^{\ell\ell}$ distributions of expected event yields for
quadratic BSM term only, showing contribution from signal and each background
processes, for coefficients $C_{G+}/\Lambda^{4}$ (left), $C_{G-}/\Lambda^{4}$
(center) and $C_{\tilde{B}W}/\Lambda^{4}$ (right).

Figure 4: $p_{\text{T}}^{\ell\ell}$ distributions of expected event yields for
interference BSM term only, showing contribution from signal and main
background process, for coefficients $C_{G+}/\Lambda^{4}$ (left),
$C_{G-}/\Lambda^{4}$ (center) and $C_{\tilde{B}W}/\Lambda^{4}$ (right).

Usually main backgrounds are estimated from data, using special control
regions, enriched by the corresponding events. Therefore, any estimation of
the background (its signal strength) already includes some BSM contributions,
and method of composite anomalous signal can become not valid. However, the
method works correctly if the control region is enriched by the SM background
events and has small sensitivity to the anomalous couplings. This can be
reached if the control region is splitted by the variable that is not
sensitive to anomalous couplings. Moreover, it is possible to use a sensitive
variable but in a non-sensitive region, e.g. $p_{\text{T}}^{\ell\ell}$ in
low-$p_{\text{T}}^{\ell\ell}$ region, that is enriched by the SM events and is
not enriched by the anomalous events, as can be seen in Fig. 2. Finally, the
method is obviously valid if the background is estimated directly from the
Monte Carlo simulation. Often these conditions are satisfied in real analyses
[10, 9, 26], so the composite anomalous signal can be used.

##  4 Statistical limit-setting procedure and results

Limits on the Wilson coefficients are set using frequentist statistical method
with likelihood-ratio-based test statistic [27], defined as

$t_{\boldsymbol{\mu}}=-2\ln\frac{L(\boldsymbol{\mu},\hat{\hat{\boldsymbol{\theta}}}(\boldsymbol{\mu}))}{L(\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\theta}})},$
(8)

where $\boldsymbol{\mu}$ and $\boldsymbol{\theta}$ are vectors of parameters
of interest and nuisance parameters respectively,
$L(\boldsymbol{\mu},\boldsymbol{\theta})$ is a likelihood function,
$(\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\theta}})$ is its global maximum and
$\hat{\hat{\boldsymbol{\theta}}}(\boldsymbol{\mu})$ maximizes the likelihood
function for a fixed $\boldsymbol{\mu}$. Confidence region for the confidence
level (CL) of 95% is defined as a region in $\boldsymbol{\mu}$-space, that
satisfies the condition

$p_{\boldsymbol{\mu}}=\int\limits_{t_{\boldsymbol{\mu}}^{\text{obs}}}^{\infty}f(t_{\boldsymbol{\mu}}|\boldsymbol{\mu})\,\text{d}t_{\boldsymbol{\mu}}>0.05,$
(9)

where $t_{\boldsymbol{\mu}}^{\text{obs}}$ is the observed value of the test
statistic and $f(t_{\boldsymbol{\mu}}|\boldsymbol{\mu})$ is the distribution
of the test statistic. The latter is assumed to be asymptotic, i.e. chi-
squared with one or two degrees of freedom for one- or two-dimensional limits
respectively, according to the Wilks’ theorem [28].

Likelihood function consists of a product of Poisson distributions for each
bin and Gaussian constraints of the nuisance parameters. For linear +
quadratic model binning as in Figs. 2 and 3 is used, whereas for linear model
binning is reduced to the one from Fig. 4. Separate nuisance parameters are
created for Monte Carlo uncertainties in each bin, and additional systematic
uncertainty of 10% is applied in order to make the limits more real, but do
not contribute significantly to the results. Since this study does not use
experimental data, SM prediction is used as data to obtain the expected
limits.

The one-dimensional limits on the Wilson coefficients in linear + quadratic
model are set for three cases: A) if only signal process BSM contributions are
accounted, B) if BSM contributions from only one background are added to the
signal one, C) if contributions from signal and all the backgrounds are
accounted simultaneously. Table 3 contains the limits for case A and impact
from each background BSM contribution on the limits. This impact shows how
much the confidence interval from the case B is stronger than the one from the
case A. Finally, the limits for the case C and impact from all the background
EFT contributions are presented. It can be seen that the main improvement
comes from ${WZ\rightarrow\ell\nu\ell\ell}$ background, whereas other
backgrounds have negligible effect on the limits on the Wilson coefficients.
The exceptions are $\mathcal{O}_{G-}$ and $\mathcal{O}_{BB}$ operators, since
${WZ\rightarrow\ell\nu\ell\ell}$ production has small and zero sensitivity to
them respectively. The most significant improvement is obtained for
coefficients $C_{G+}/\Lambda^{4}$ and $C_{WW}/\Lambda^{4}$ due to the large
sensitivity of ${WZ\rightarrow\ell\nu\ell\ell}$ production to the new physics
described by corresponding operators.

Table 3: Impact of the BSM background contributions on the one-dimensional limits on six Wilson coefficients in linear + quadratic model. Coef. |  Limits [TeV-4], | Impact from separate bkgs. |  Limits [TeV-4], | Total bkg. impact
---|---|---|---|---
sig.-only EFT | ${WZ\rightarrow\ell\nu\ell\ell}$ | Non-res. | ${Z(\rightarrow\ell\ell)\text{ + jets}}$ | ${ZZ\rightarrow 4\ell}$ | all bkgs. EFT
$C_{G+}/\Lambda^{4}$ | [-0.122; 0.122] | 57.5% | 0 | 8.2% | 0.7% | [-0.051; 0.050] | 58.8%
$C_{G-}/\Lambda^{4}$ | [-0.422; 0.423] | 1.6% | 0.2% | 1.0% | 0.6% | [-0.408; 0.410] | 3.3%
$C_{\tilde{B}W}/\Lambda^{4}$ | [-0.679; 0.686] | 2.8% | 0 | 0.2% | 0.5% | [-0.654; 0.665] | 3.4%
$C_{BW}/\Lambda^{4}$ | [-1.57; 1.55] | 3.4% | 0 | 0.3% | 0.5% | [-1.50; 1.49] | 4.1%
$C_{BB}/\Lambda^{4}$ | [-0.836; 0.839] | 0 | 0 | 0.2% | 0.6% | [-0.829; 0.833] | 0.8%
$C_{WW}/\Lambda^{4}$ | [-1.31; 1.28] | 11.1% | 0 | 0.5% | 0.5% | [-1.15; 1.13] | 11.7%

Table 4 presents the one-dimensional limits on the Wilson coefficients of
$CP$-conserving operators in linear model. They were set without accounting
for background BSM contributions and with accounting ones from
${WZ\rightarrow\ell\nu\ell\ell}$ background. Interference terms of other
backgrounds are very small and, therefore, are not taken into account. The
limits in linear model are much weaker than the ones in linear + quadratic
model, since at the current experimental sensitivity quadratic term dominates.
However, the limits can be significantly improved by composite anomalous
signal, background impact on the limits is also presented in Table 4. Note
that in general case limits in linear model can be improved as well as
worsened. For example, interference terms of ${ZZ\rightarrow\ell\ell\nu\nu}$
and ${WZ\rightarrow\ell\nu\ell\ell}$ productions for $\mathcal{O}_{G-}$
operator have different signs, it can be seen in Fig. 4. Therefore, usage of
different binning and selection compared to the ones in this work can lead to
the worsening of the limits.

Table 4: Impact of the BSM background contributions on the one-dimensional limits on six Wilson coefficients in linear model. Coef. | Limits [TeV-4], sig.-only EFT | Limits [TeV-4], sig.+bkg. EFT | Bkg. impact
---|---|---|---
$C_{G+}/\Lambda^{4}$ | [-147; 229] | [-9; 11] | 94.4%
$C_{G-}/\Lambda^{4}$ | [-110; 94] | [-57; 60] | 42.9%
$C_{\tilde{B}W}/\Lambda^{4}$ | [-77.5; 66.2] | [-37; 32] | 56.2%

Additionally, two-dimensional limits are also set with and without BSM
contributions from ${WZ\rightarrow\ell\nu\ell\ell}$ background, dropping BSM
contributions from other backgrounds due to their smallness. $CP$-even and
$CP$-odd operators have zero interference (cross-term) in this study. Thus,
two-dimensional limits are set on six pairs of the Wilson coefficients, and
they are presented in Fig. 5. As it was mentioned above, for some coefficient
pairs cross-term leads to the change of the sensitivity compared to the one-
dimensional limits. Area outside the presented contours is excluded at 95% CL.
The improvement of the two-dimensional limits due to taking into account
${WZ\rightarrow\ell\nu\ell\ell}$ BSM contributions varies from 6.7% for
$C_{BW}/\Lambda^{4}$ vs $C_{BB}/\Lambda^{4}$ contour to 59.1% for
$C_{G+}/\Lambda^{4}$ vs $C_{\tilde{B}W}/\Lambda^{4}$ contour.

Figure 5: Two-dimensional 95% CL limits on six pairs of the Wilson
coefficients. Blue (red) contours were obtained under usage of the anomalous
contributions from ${ZZ\rightarrow\ell\ell\nu\nu}$
(${ZZ\rightarrow\ell\ell\nu\nu}+{WZ\rightarrow\ell\nu\ell\ell}$) production.

##  5 Conclusion

This paper presents a study of BSM contributions of background processes in
the search of anomalous neutral triple gauge couplings using
${ZZ\rightarrow\ell\ell\nu\nu}$ production in $pp$ collisions at $\sqrt{s}=13$
TeV with a dataset of 140 fb-1. Six dimension-eight operators describing
corresponding couplings were considered. The results obtained in linear +
quadratic model showed that, generally, only ${WZ\rightarrow\ell\nu\ell\ell}$
background changes the limits on the Wilson coefficients significantly, and
the most significant improvement was found for $C_{G+}/\Lambda^{4}$ (57.5%)
and $C_{WW}/\Lambda^{4}$ (11.1%) coefficients. Additionally, the improvement
of the limits on the Wilson coefficients in linear model was derived and is of
42.9%-94.4% depending on the operator. Finally, two-dimensional limits in
linear + quadratic model were considered, and the confidence regions became
tighter up to 59.1%.

Wilson coefficients can be turned into the model-dependent parameters.
Therefore, experimental limits on these coefficients allow constraining some
parameters of BSM theories. So, different possibilities of improvement of the
limits on the Wilson coefficients, like the presented in this work method of
composite anomalous signal, should be studied and used in the analyses of the
experimental data.

## Acknowledgements

This work was supported by the Russian Science Foundation under grant
21-72-10113.

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