# A Comparison of Methods for Neural Network Aggregation

John Pomerat Aviv Segev Department of Computer Science, University of South
Alabama, Mobile, AL, 36688 USA e-mail<EMAIL_ADDRESS>
###### Abstract

Deep learning has been successful in the theoretical aspect. For deep learning
to succeed in industry, we need to have algorithms capable of handling many
inconsistencies appearing in real data. These inconsistencies can have large
effects on the implementation of a deep learning algorithm. Artificial
Intelligence is currently changing the medical industry. However, receiving
authorization to use medical data for training machine learning algorithms is
a huge hurdle. A possible solution is sharing the data without sharing the
patient information. We propose a multi-party computation protocol for the
deep learning algorithm. The protocol enables to conserve both the privacy and
the security of the training data. Three approaches of neural networks
assembly are analyzed: transfer learning, average ensemble learning, and
series network learning. The results are compared to approaches based on data-
sharing in different experiments. We analyze the security issues of the
proposed protocol. Although the analysis is based on medical data, the results
of multi-party computation of machine learning training are theoretical and
can be implemented in multiple research areas.

##  1 Introduction

In recent years, the theoretical progress of machine learning promises to
revolutionize many domains of industry, from manufacturing[1], through
healthcare [2] and transportation[19], to education[20]. Although there have
been many successful implementations of learning algorithms, much of the
progress in machine learning remains theoretical [3]. One reason for the lack
of implementation, particularly in the healthcare domain, is the practicality,
resilience, and security of learning algorithms [2, 4]. A staple of machine
learning is data; as such, its shape, organization, quantity, and quality must
all be carefully considered for many real-world implementations [5]. As the
need for healthcare datasets rises, data-sharing [5] has been suggested as a
strategy to get data for healthcare models. In data-sharing, hospitals
reformat data into an agreed upon structure and anonymize contents so as not
to expose confidential patient data. We propose an alternative to data-sharing
using secure multi-party computation (MPC). Multi-party computation is a
branch of cryptography concerned with calculating functions on private, user-
held data. One motivating example considers two people who wish to determine
which of them has a higher salary without either party exposing their salary
to one-another. With MPC, there is an algorithm capable of solving this
problem and other, similar, problems. We propose a protocol for training
neural networks on private datasets then combining the neural networks such
that private data is not exposed, and the model’s final performance is
comparable to a model trained on the combined private datasets. This paper
considers three methods of neural network aggregation to combine networks
trained on distinct datasets sharing an underlying function. For all three of
the methods, underlying network architecture, datasets were kept constant.
Additionally, hyperparameters, including activation functions, optimizer, and
batch size were held constant. Performance was measured by mean square error,
which was recorded to compare the three methods. The three methods are
transfer learning, average ensemble learning, and series network learning.

This paper will explore these methods of neural network aggregation in depth.

##  2 Related Work

###  2.1 Security

In the healthcare domain, the importance of maintaining data privacy is clear.
As such, sensitive data should be anonymized as much as possible to prevent
any kind of data leakage. There are a number of attacks against learning
algorithms [6, 7, 8, 9]. In 2015, Goodfellow et al. proposed adversarial
attacks as a security vulnerability of neural networks [10]. Since then, there
has been more research into the security of neural networks and more attack
vectors have been discovered [6, 7, 8, 9]. Our problem, as we have defined it,
is not vulnerable to black box adversarial attacks. One attack vector that our
system is vulnerable to is training code provided by a malicious adversary
[11]. To protect against this attack, the code which specific parties
implement should be open-sourced and independently reviewed. Additionally,
there is an attack vector for generative models[12] which should be considered
for some implementations with generative models but is beyond the scope of
this paper. The primary attack vector of concern is the membership inference
attack [18]. The membership inference attack is a blackbox attack vector for a
trained neural network classifier. The attack is an algorithm to statistically
determine from a trained neural network whether an input tuple is a member of
the private training set or not[18]. To protect against the membership
inference attack, models should avoid overfitting. Additionally, adding
regularization, prediction vector restrictions, and increasing the entropy of
the prediction vector have value in preventing membership inference attacks
[18].

###  2.2 Transfer Learning

A well known method of neural network aggregation is transfer learning.
Recently, transfer learning has been shown to be useful and extremely
versatile, particularly with reinforcement learning and deep neural network
models[22, 23, 16, 24]. Additionally, transfer learning is also more versatile
than some of the methods explored in this paper since it is capable of working
with a wider variety of learning algorithms including convolutional neural
networks[22]. Furthermore, transfer learning has been shown to work well with
time series predictions and recurrent neural networks [25]. In addition,
research on transfer learning in the context of the healthcare domain already
shows promise [26, 27, 28]. One paper by Gupta et al.[26] leveraged transfer
learning to generalize models in the healthcare domain to similar tasks in the
same domain. Similarly, a result by Chen et al.[27] with wearable technology
used transfer learning on time series health data to improve performance and
increase personalization of the FedHealth model. The results for transfer
learning show promise for the viability of neural network aggregation for deep
learning in the healthcare domain.
Previous work in transfer learning shows great promise for neural network
aggregation as an alternative to data-sharing in big data healthcare
applications. These results, combined with some of the research conducted in
security and multi-party computation act as the motivating examples for this
paper.

##  3 Neural Network Aggregation

###  3.1 Problem Statement

The setup goes as follows. Let $D_{1},D_{2},\cdots,D_{k}$ be subsets of
$\mathbb{R}^{n}$ represented as datasets. Then, let
$G:\mathbb{R}^{n}\to\mathbb{R}^{m}$ be a differentiable function represented
as a multilayer perceptron with parameters $\theta_{g}$. We are concerned with
methods of producing $\theta_{g}$ from the $D_{i}$ such that the loss of $G$
is comparable to obtaining the $\theta_{g}$ from $\bigcap_{i=1}^{k}D_{i}$ and
it is not computationally feasible to extract information about the members of
the $D_{i}$ from $G$. This process, of training a neural network from
multiple, disjoint datasets is called neural network aggregation. The three
methods of neural network aggregation are series network learning, average
ensemble learning, and transfer learning.

###  3.2 Series Network Learning

The first method, called series network learning, functions by training a
neural network with a pretrained “expert” neural network as an additional
input. For our experiment we consider a single neural network trained on the
first dataset. The neural network’s performance on the testing set is
recorded. The network then generates a prediction for every entry in the
second dataset, a new neural network is then created for the second dataset
with the prediction array as an additional input. The neural network is then
trained on the second dataset and the mean square error is recorded.
Intuitively, the second neural network will likely have an improvement in mean
square error as the network will “learn” when to trust the first network’s
prediction and when to instead use its own calculations, Fig. 1.

Figure 1: Neural network as an input to assist training a second network.

for  _For all parties except the last_ do

train network on parties data;

end for

take network and append each output of the trained networks as a new input
neuron then train resulting network on the final parties data;

Algorithm 1 Series Networks

###  3.3 Average Ensemble Learning

The second method considers two neural networks, $N_{1},N_{2}$ of the exact
same architecture, with the same activation functions, optimizer, number of
hidden layers, and number of neurons. Each of the networks are then trained on
different datasets of identical structure and mean square error on the testing
set is recorded. Then, the two neural networks are combined to form a third
network of the exact same structure $N_{3}$ (Fig. 2). The weights and biases
of $N_{3}$ are the average of the corresponding weights and biases in $N_{1}$
and $N_{2}$. More specifically, if $n$ is the total number of weights and
biases in $N_{3}$, and $N_{3}(i)$ refers to the $i$-th weight or bias in
$N_{3}$, then for all $0<i\leq n$,

$N_{3}(i)=\frac{N_{1}(i)+N_{2}(i)}{2}$

$N_{3}$ is then measured on the testing set and its performance is compared to
the performance of both $N_{1}$ and $N_{2}$. In addition to a pure average,
other strategies are considered. Initially, a weighted average may be
performed with weights proportional to the size of the dataset to guarantee
that a model trained on significantly more data is not treated the same as a
model trained on a much smaller set of data. Another option is to use a
weighted average not only with the size of the dataset, but also the ratio of
positives and negatives for disease prediction cases. This is done to ensure
that a larger dataset, which is not highly informative, will not overpower a
smaller dataset containing more information.

Figure 2: Averaging the corresponding weights and biases of a neural network

for  _For all parties_ do

train identical network on parties data;

end for

initialize new model identical to the others;

for  _every weight and bias in the network_ do

for  _every trained network_ do

sum values of corresponding weight or bias;

end for

weight or bias in new network is that sum divided by the number of parties;

end for

Algorithm 2 Average Ensemble Learning

###  3.4 Transfer Learning

The third method is transfer learning. Instead of combining two neural
networks, transfer learning functions by training on additional datasets with
a single neural network without weight reinitialization[16, 17]. Our
experiment considers a single neural network with randomly initialized weights
trained on the first dataset. The mean squared error on the testing set is
recorded. Then, the neural network is trained on the second dataset without
reinitializing the weights and the mean squared error is recorded again. This
process is then repeated by training on the second dataset and then the first.
With mean square error being recorded throughout.

##  4 Experiments

To compare the proposed methods of neural network aggregation, we ran two
experiments, one with artificially generated polynomial data, and the other on
the University of Wisconsin Madison Hospital’s breast cancer dataset[21]. The
motivation of these tests was to get an initial performance comparison between
the proposed methods and a neural network trained on all of the data
simultaneously representing data-sharing.

###  4.1 Data

The neural networks in this paper were trained on both real and artificially
generated data. The artificially generated data was created as follows. A
random normal distribution was employed to create 2 dimensional arrays
populated with random rational numbers in a specified range. The rows of the
array consisted of 7 random rational numbers representing data features.
Multiple datasets were created for the experiment. Arrays of size 3200, 1600,
800, and 400 were created. After the arrays were generated, a multivariate
polynomial of degree $n$ under lexicographic term ordering was created with
coefficients randomly chosen from a normal distribution.

$f(x_{1},x_{2}...x_{7})$ (1)

Next, for each set of $7$ values in the generated data, $\gamma$, $f(\gamma)$
was calculated by plugging the values from the generated data into the
polynomial Fig. 3 illustrates this in $2$ dimensions as opposed to $7$.

Figure 3: Random x values (red) and calculated y values (green) on the
generated polynomial function (blue).

After $f(\gamma)$ has been calculated for all the tuples in each array, the
values were combined with the generated data to form a dataset such that each
row contains 8 values, 7 random rational numbers, and the calculated y-value
according to the generated function. Thus, the networks in the experiments
will be trained on the 7 rational numbers to learn the underlying polynomial
function. These datasets were then divided into two training sets and a
testing set containing 80% and 20% of the entries respectively. The training
set was then divided again into two training sets of equal size.

The real data used in this paper comes from the University of Wisconsin
Madison Hospital’s breast cancer dataset[21]. This dataset was also divided
into two equally sized training sets and a single testing set. The breast
cancer dataset contains 569 rows and 32 features. This was split into two
training sets each with 256 training examples and a testing set with 57
examples. The features in the breast cancer data describe tumors. Some of the
features include clump thickness, uniformity of cell size and shape, marginal
adhesion, and others. Furthermore, each of these features was recorded in
three different ways in the data. For each feature, an average, a low, and a
high value were all available in the data. The data preprocessing used
consisted of normalization and minor feature manipulations to get the data in
the right shape to form proper training and testing sets.

###  4.2 Regression

For this experiment, the regression data (as defined above) was taken, then
split into a training set and a testing set with $80\%$ of the examples for
training and $20\%$ of the examples for testing. A neural network was trained
on the training set, then loss on testing set was recorded. Since all the data
was in one place, the resulting model represents a network trained on a
dataset created through data-sharing. Then, the training set was split into
two smaller training sets of equal size. Then, we perform each of the three
methods to train a neural network from the split datasets recording loss for
each. The neural network architecture was chosen to best fit the data and was
kept constant while the test was repeated many times under different
conditions. The conditions varied epochs from 10 to 200, noise in the
regression data from a logarithmic shift with coefficients varying between 1
and 3, size of the datasets from 400 to 32,000, and polynomial degree of the
underlying dataset from 2 to 5. The average loss for all methods, including
the loss from the data-sharing model represented as “None”, across all tests
can be seen in below in Fig. 4. Additionally, average loss for tests with
varying degrees of added noise can be found in Table 1. The added noise in the
data is given by

$y=f(x_{1},x_{2},...x_{7})+ndr$

where $x_{1},x_{2}...x_{7}$ is a data point, $n$ a chosen noise value, $r$ is
a random real number selected from a random normal distribution between -2 and
2, and $f$ is a polynomial function with degree $d$.

Figure 4: Loss comparison of methods on regression data Method | Average MSE | Noise n
---|---|---
| 0.015 | 0
Average Ensemble | 0.011 | 1
| 0.011 | 2
| 0.013 | 0
Series Networks | 0.010 | 1
| 0.010 | 2
| 0.011 | 0
Transfer Learning | 0.007 | 1
| 0.009 | 2
| 0.006 | 0
None | 0.008 | 1
| 0.008 | 2
Table 1: Loss comparison for methods with added noise

Preperformance is the loss measured on the testing set once the model had
learned on the first dataset. Similarly, postperformance is the loss measured
after the second dataset had been aggregated in. The purpose of this is to see
the method converge to the performance of the model obtained through data-
sharing by aggregating in multiple datasets. After training, all three methods
achieved comparable aggregate performance compared to the model trained on the
combined “shared” data (None). Here, series networks had the best performance
of the three methods and also had the greatest performance increase after
aggregation.

###  4.3 Breast Cancer Classification

For this experiment, the goal is to train a classifier to determine whether a
tumor is benign or malignant. The breast cancer dataset contains 569 rows and
32 features. Similarly to the regression experiment, the data was split, a
neural network architecture was configured for the data, then accuracy values
for the three methods were computed. Additionally, the data-sharing equivalent
model was trained on the data before the training sets were bifurcated and the
accuracy was recorded. Tests were repeated with varied hyperparameters,
including, batch size, epochs, and number of neurons. The accuracy values of
the test can be found in Fig. 5, the ROC curve for the test is in Fig. 6, and
precision, recall, and F1 scores are in Table 2.

Here, all of the methods performed better than the equivalent model obtained
through data-sharing. Additionally, this example also provides evidence for
the viability of our method in the healthcare domain.

Figure 5: Accuracy comparison of methods on breast cancer data Figure 6: ROC Curve for Breast Cancer Data Method | Precision | Recall | F1 Score
---|---|---|---
Average Ensemble | 0.76 | 1.00 | 0.87
Series Networks | 1.00 | 0.93 | 0.96
Transfer Learning | 1.00 | 0.93 | 0.96
Table 2: Metrics for compared methods

From the accuracy graph (Fig. 5) and the ROC curve (Fig. 6), transfer learning
and series networks performed the best, outperforming training on the combined
dataset. This is likely due to the fact that with smaller dataset size,
training on smaller subsets of the data grants more generalization.

##  5 Conclusion

In order for neural network aggregation to be fully recognized as a stronger
alternative to data-sharing, more tests need to be run. Additionally, future
work should examine the scaling of the proposed model, examining for model
convergence as the number of disjoint datasets increases. If transfer learning
or series network learning is able to converge to the same model acquired
through data-sharing by distributing training across many datasets, then the
method would be viable. Furthermore, more studies need to be conducted on
membership inference attacks to lower the security concerns. Since the
membership inference attack is strong against overfit models, it would be
interesting to see what the end behavior of series networks or transfer
learning after training on many datasets. Despite this, both transfer learning
and series network learning seem to be promising methods for distributing
training on private datasets. Thus, while more tests need to be conducted to
prove the viability of neural network aggregation, this paper establishes an
initial claim for neural network aggregation as a functional alternative to
data-sharing.

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# Rare Kaon Decay to Missing Energy: Implications of the NA62 Result for a
$Z^{\prime}$ Model

Téssio B. de Melo1<EMAIL_ADDRESS>Sergey Kovalenko2
<EMAIL_ADDRESS>Farinaldo S. Queiroz1,3<EMAIL_ADDRESS>C. Siqueira1,4<EMAIL_ADDRESS>Yoxara S. Villamizar1,3
<EMAIL_ADDRESS>1 International Institute of Physics, Universidade Federal
do Rio Grande do Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970,
Brazil
2 Departamento de Ciencias Físicas, Universidad Andres Bello, Sazie 2212,
Santiago, Chile
3 Departamento de Física, Universidade Federal do Rio Grande do Norte,
59078-970, Natal, RN, Brasil
4 Instituto de Física de São Carlos, Universidade de São Paulo, Av.
Trabalhador São-carlense 400, São Carlos, Brasil.

###### Abstract

Meson decays offer a good opportunity to probe new physics. The rare kaon
decay $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ is one of the cleanest of them
and, for this reason, is rather sensitive to new physics, in particular,
vector mediators. NA62 collaboration, running a fixed-target experiment at
CERN, recently reported an unprecedented sensitivity to this decay, namely a
branching fraction of
$BR(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=(11^{+4.0}_{-3.5})\times 10^{-11}$ at
68% C.L. Vector mediators that couple to neutrinos may yield a sizeable
contribution to this decay. Motivated by the new measurement, we interpret
this result in the context of a concrete $Z^{\prime}$ model, and put our
findings into perspective with the correlated
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ decay measured by KOTO collaboration,
current, and future colliders, namely the High-Luminosity and High-Energy LHC.

##  I Introduction

Mesons have played a key role in the understanding of properties of elementary
particles. The introduction of strangeness along with isospin lead us to the
eight-fold way, based on the SU(3) flavor symmetry. These theoretical insights
have contributed to the discovery of quantum chromodynamics as we know today.
Another good example is the famous $\theta-\tau$ puzzle. Two different decays
were found for charged strange mesons known at the time as $\theta^{+}$ and
$\tau^{+}$. The decay modes had different parities, but the particles were
shown to have the same mass and lifetime. It was indeed a puzzling
observation. Later, it was realized that weak interactions violate parity, and
these two particles were actually the same $K^{+}$-meson. Additionally, the
Glashow-Iliopoulos-Maiani (GIM) mechanism and quark charm surfaced as an
explanation of the absence of weak flavor changing neutral currents in the
processes such as $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$. The discovery of CP
violation in the $K^{0}-\bar{K}^{0}$ system further proved the importance of
meson physics for our understanding of nature. Furthermore, meson systems are
able to access possible new sources of CP violation that are of paramount
importance for explaining the observed baryon-antibaryon asymmetry in our
universe [1]. Lastly, the $K^{+}$ rare decay into neutrinos can efficiently
probe the presence of heavy vector mediators, beyond the Standard Model (SM),
[2, 3, 4, 5, 6, 7, 8, 9] via the Feynman diagrams displayed in Fig.1.

Figure 1: Feynman diagrams that illustrate how vector mediators can contribute
to the $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ decay. The first diagram requires
$Z^{\prime}$ coupling to neutrinos, whereas the second further requires
couplings to the top quark.

The meson decay $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ is a flavor changing
neutral current process that occurs in the SM via the box and penguin
diagrams, with the latter being dominated by the top quark contribution. Due
to the GIM and loop suppression, the SM contribution to this decay is very
small, reading ${\rm BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=(8.4\pm
1.0)\times 10^{-11}$ [10], while the NA62 currently imposes ${\rm
BR}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=11.0^{+4.0}_{-3.5}\times 10^{-11}$
[11, 12] (results collected in 2016, 2017 and 2018). Therefore, one can
clearly notice that the current sensitivity achieved by NA62 is not far from
the SM prediction. Having said that, the NA62 collaboration has been
continuously searching for the rare kaon decay [12]. KOTO collaboration has
also conducted some searches as well, but offering weaker constraints
[13]111KOTO collaboration has also recently reported the observation of three
anomalous events in the $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$. This anomaly
requires the branching ratio for this decay mode to be about two orders of
magnitude larger than the SM one [4], indicating the presence of a new light
and long-lived particle with mass of the order of $100$ MeV. There is no such
light particle in our model. Hence the KOTO anomaly will be regarded as an
statistical fluke.. To concretely show the relevance of the recent NA62
result, we will put it into perspective other existing limits in a model that
features a heavy vector mediator. The model is based on the $SU(3)_{C}\times
SU(3)_{L}\times U(1)_{Y}$ gauge group, known as 3-3-1 model. It is well
motivated by the ability to naturally explain the observed replication of the
generations, and nicely hosting dark matter candidates [14, 15, 16, 17, 18,
19, 20, 21], addressing neutrino masses [22, 23, 24, 25], among other
interesting phenomena [26, 26, 27, 3, 28, 29, 30, 31, 32, 33]. As a result of
the enlarged gauge group, the 3-3-1 model have several new gauge bosons, such
as $W^{\prime}$ and $Z^{\prime}$ bosons, which are subject to restrictive
constraints rising from collider physics [34, 35, 36, 37], muon anomalous
magnetic moment [38, 39, 40, 41], and lepton flavor violation [42, 43]. We
will investigate the role of the $Z^{\prime}$ gauge boson in the rate $K^{+}$
decay and then use this to draw bounds on the $Z^{\prime}$ mass using the
$K_{L}\rightarrow\pi^{0}\bar{\nu}\nu$, $K^{+}\rightarrow\pi^{+}\bar{\nu}\nu$
decays.

Our work is structured as follows: In Section II, we review the 3-3-1 model
and compute the $Z^{\prime}$ couplings necessary to our analyses; in Section
III, we discuss the computation of the branching fractions of the kaons in our
model; in Section IV, we discuss our results and, finally in Section VI, we
draw our conclusions.

##  II The Model

The 3-3-1 models are extensions of the standard model and are based on the
following local symmetry group:
$\textbf{SU(3)}_{\textbf{C}}\times\textbf{SU(3)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{N}}$,
where C corresponds to the color charge, L denotes the left-handed fermions
and N is the quantum number of the U(1) group. The general expression for the
electric charge operator in these models is written as,

$\frac{Q}{e}=\frac{1}{2}\left(\lambda_{3}+\beta\lambda_{8}\right)+\text{N}\,\text{I}=\left(\begin{array}[]{c}\frac{1}{3}+\text{N}\\\
-\frac{2}{3}+\text{N}\\\ \frac{1}{3}+\text{N}\end{array}\right),$ (1)

where $\lambda_{3}=\operatorname{diag}(1,-1,0)$,
$\lambda_{8}=\operatorname{diag}(1,1,-2)/\sqrt{3}$, and I are the diagonal
Gell-Mann matrices and the identity matrix, respectively. We took
$\beta=-\frac{1}{\sqrt{3}}$ because in our work we choose the model known as
3-3-1 with right-handed neutrinos (3-3-1 r.h.n) [44, 45]. However, we
highlight that our conclusions are also applicable to the 3-3-1 model with
neutral fermions proposed in [14]. The hypercharge in this model is given as,

$Y=2Q-\lambda_{3}=2N-\frac{\sqrt{3}\lambda_{8}}{3},$ (2)

which is identical to the standard model one. The left (L) and right
(R)-handed fermionic fields of this model are represented as follows,

$\begin{array}[]{c}f_{L}^{a}=\left(\begin{array}[]{c}\nu_{L}^{a}\\\
\ell_{L}^{a}\\\
\left(\nu_{R}^{c}\right)^{a}\end{array}\right)\sim(1,3,-1/3),\\\ \\\
\ell_{R}^{a}\sim(1,1,-1),\end{array}$ (3)

$\begin{array}[]{c}Q_{iL}=\left(\begin{array}[]{c}d_{iL}\\\ -u_{iL}\\\
d_{iL}^{\prime}\end{array}\right)\sim(3,\overline{3},0),\\\ \\\
u_{iR}\sim(3,1,2/3),\,d_{iR}\sim(3,1,-1/3),\\\ \\\
d_{iR}^{\prime}\sim(3,1,-1/3),\\\ \\\ Q_{3L}=\left(\begin{array}[]{c}u_{3L}\\\
d_{3L}\\\ T_{L}\end{array}\right)\sim(3,3,1/3),\\\ \\\
u_{3R}\sim(3,1,2/3),\,d_{3R}\sim(3,1,-1/3),\\\ \\\
T_{R}\sim(3,1,2/3),\end{array}$ (4)

where $a=1,2,3$ and $i=1,2$ are the generation indexes, $f^{a}_{L}$ and
$Q_{iL}$, $Q_{3L}$ represent the left-handed leptonic and quark triplets,
respectively. These fields encompass the SM spectrum like neutrinos
($\nu^{a}=\nu_{e},\nu_{\mu},\nu_{\tau}$), charged leptons
$\ell^{a}=e,\mu,\tau$, and quarks $u_{i}=\overline{u},\overline{c}$,
$d_{i}=\overline{d},\overline{s}$, $u_{3}=t$ and $d_{3}=b$. Besides, there are
other particles additional to the SM: the right-handed neutrino
$\left(\nu_{R}^{c}\right)^{a}$ and three new heavy exotic quarks
$d_{iL}^{\prime}$ and $T_{L}$. In Eqs. (3), (4), we have specified the field
assignments indicating how they transform under the symmetries
$\left(\mathrm{SU}(3)_{c},\mathrm{SU}(3)_{L},\mathrm{U}(1)_{N}\right)$,
respectively. The values of their electric charge and hypercharge can be found
from Eqs. (1) and (2).

Furthermore, the 3-3-1 r.h.n model contains three scalar fields $\chi$, $\eta$
and $\rho$ in the following representations

$\displaystyle\quad\quad\chi=\left(\begin{array}[]{c}\chi^{0}\\\ \chi^{-}\\\
\chi^{\prime 0}\end{array}\right)\sim(1,3,-1/3),$ (8)
$\displaystyle\rho=\left(\begin{array}[]{c}\rho^{+}\\\ \rho^{0}\\\
\rho^{\prime+}\end{array}\right)\sim(1,3,2/3),$ (12)
$\displaystyle\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}\\\
\eta^{\prime 0}\end{array}\right)\sim(1,3,-1/3).$ (16)

These scalar triplets in Eq. (16) are responsible for the spontaneous symmetry
breaking (SSB) in the model, with the following vacuum (vev) structure,

$\langle\chi\rangle=\left(\begin{array}[]{c}0\\\ 0\\\
v_{\chi}\end{array}\right),\langle\rho\rangle=\left(\begin{array}[]{c}0\\\
v_{\rho}\\\
0\end{array}\right),\langle\eta\rangle=\left(\begin{array}[]{c}v_{\eta}\\\
0\\\ 0\end{array}\right),$ (17)

.

We will assume that $v_{\chi}\gg v_{\eta},v_{\rho}$, leading to the two-step
SSB,

$\displaystyle\textbf{SU(3)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{X}}\xrightarrow{\quad\langle\chi\rangle\quad}\textbf{SU(2)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{Y}}\xrightarrow{\langle\eta\rangle,\langle\rho\rangle}\textbf{U(1)}_{\textbf{Q}}$

with $U(1)_{Q}$, being the $U(1)$ from electrodynamics.

The fermion masses rise from the Yukawa Lagrangian that reads,

$\displaystyle\mathcal{L}_{Yuk}$
$\displaystyle=\lambda_{1a}\bar{Q}_{1L}d_{aR}\rho+\lambda_{2ia}\bar{Q}_{iL}u_{aR}\rho^{*}+G_{ab}^{\prime}\bar{f}_{L}^{a}e_{R}^{b}\rho$
(18)
$\displaystyle+G_{ab}\varepsilon^{ijk}\left(\bar{f}_{L}^{a}\right)_{i}\left(f_{L}^{b}\right)_{j}^{c}\left(\rho^{*}\right)_{k}+\lambda_{3a}\bar{Q}_{1L}u_{aR}\eta$
$\displaystyle+\lambda_{4ia}\bar{Q}_{iL}d_{aR}\eta^{*}+\lambda_{1}\bar{Q}_{3L}T_{R}\chi+\lambda_{2ij}\bar{Q}_{iL}d_{jR}^{\prime}\chi^{*}+H.c.$

The quark and charged lepton masses are proportional to $v=246$ GeV, where
$v^{2}=v_{\rho}^{2}+v_{\eta}^{2}$ similarly to the SM. The fourth term leads
to a $3\times 3$ antisymmetric neutrino mass matrix [45], which means that the
model has one massless and two degenerate neutrino mass eigenstates. Moreover,
the gauge bosons $W$ and $Z$ acquire mass term identical to the SM as well. In
addition to the SM fields, there are new massive gauge bosons predicted by the
model as result of the enlarged gauge symmetry, denoted as
$Z^{\prime},V^{\pm}$ and $U^{0},U^{0\dagger}$. The masses of these fields are,

$\displaystyle M_{W^{\pm}}^{2}$
$\displaystyle=\frac{1}{4}g^{2}v^{2},M_{Z}^{2}=\frac{M_{W^{\pm}}^{2}}{C_{W}^{2}}$
(19) $\displaystyle M_{Z^{\prime}}^{2}$
$\displaystyle=\frac{g^{2}}{4\left(3-4S_{W}^{2}\right)}\left[4C_{W}^{2}v_{\chi}^{2}+\frac{v^{2}}{C_{W}^{2}}+\frac{v^{2}\left(1-2S_{W}^{2}\right)^{2}}{C_{W}^{2}}\right]$
$\displaystyle M_{V^{\pm}}^{2}$
$\displaystyle=\frac{1}{4}g^{2}\left(v_{\chi}^{2}+v^{2}\right),M_{U^{0}}^{2}=\frac{1}{4}g^{2}\left(v_{\chi}^{2}+v^{2}\right).$

with $M_{W}\ll M_{U},M_{V}$, $S_{W}=\sin\theta_{W}$ and
$C_{W}=\cos\theta_{W}$, with $\theta_{W}$, the Weinberg angle.

The charged (CC) and neutral (NC) currents are found to be,

$\displaystyle\mathcal{L}^{CC}=$
$\displaystyle-\frac{g}{\sqrt{2}}\left[\bar{\nu}_{L}^{a}\gamma^{\mu}e_{L}^{a}W_{\mu}^{+}+\left(\nu_{R}^{\bar{c}}\right)^{a}\gamma^{\mu}e_{L}^{a}V_{\mu}^{+}\right.$
(20)
$\displaystyle+\left.\bar{\nu}_{L}^{a}\gamma^{\mu}\left(\nu_{R}^{c}\right)^{a}U_{\mu}^{0}\right]$
$\displaystyle-\frac{g}{\sqrt{2}}\left[\left(\bar{u}_{3L}\gamma^{\mu}d_{3L}+\bar{u}_{iL}\gamma^{\mu}d_{iL}\right)W_{\mu}^{+}\right.$
$\displaystyle+\left.\left(\bar{T}_{L}\gamma^{\mu}d_{3L}+\bar{u}_{iL}\gamma^{\mu}d_{iL}^{\prime}\right)V_{\mu}^{+}\right.$
$\displaystyle+\left(\bar{u}_{3L}\gamma^{\mu}T_{L}-\bar{d}_{iL}^{\prime}\gamma^{\mu}d_{iL}\right)U_{\mu}^{0}+\text{
h.c. }],$

$\displaystyle\mathcal{L}^{NC}$
$\displaystyle=\frac{g}{2c_{W}}\left\\{\bar{f}\gamma^{\mu}\left[a_{1L}(f)\left(1-\gamma_{5}\right)+a_{1R}(f)\left(1+\gamma_{5}\right)\right]fZ_{\mu}^{1}\right.$
(21)
$\displaystyle\left.+\bar{f}\gamma^{\mu}\left[a_{2L}(f)\left(1-\gamma_{5}\right)+a_{2R}(f)\left(1+\gamma_{5}\right)\right]fZ_{\mu}^{2}\right\\},$

The second and third term in Eq. (20) violate leptonic number and weak isospin
[45]. $Z^{1}$ and $Z^{2}$ are neutral physical gauge bosons, which rise from
the $Z$ and $Z^{\prime}$ gauge boson mixtures. $a_{1R}(f)$, $a_{1L}(f)$,
$a_{2R}(f)$ and $a_{2L}(f)$ are couplings of fermions with the $Z^{1}$ and
$Z^{2}$ bosons. The mixing angle of these bosons is commonly denoted as $\phi$
and when $\phi=0$, the couplings of $Z^{1}$ with the leptons and quarks are
the same as the boson $Z$ in the SM. Likewise, the couplings of $Z^{2}$ in
this limiting case should be the same as $Z^{\prime}$ [45]. These couplings
for the vertices $Z^{\prime}-\nu-\overline{\nu}$ ,
$Z^{\prime}-\overline{d_{i}}-d_{i}$ and $Z^{\prime}-\overline{b}-b$ are shown
in Table 1.

| $Z^{\prime}-\nu-\overline{\nu}$ | $Z^{\prime}-\overline{d_{i}}-d_{i}$ | $Z^{\prime}-\overline{b}-b$
---|---|---|---
Coupling constant | $\frac{1-2S_{W}^{2}}{2\sqrt{3-4S_{W}^{2}}}$ | $-\frac{\sqrt{3-4S_{W}^{2}}}{6}$ | $\frac{3-2S_{W}^{2}}{6\sqrt{3-4S_{W}^{2}}}$
Table 1: $Z^{\prime}$ couplings to neutrinos and the left-handed down-type
quarks, considering $\phi=0$.

In order to study meson physics in our model, and investigate its connection
to the $Z^{\prime}$ boson, we need to extract flavor changing neutral current.
To do so, we start by reviewing how the flavor changing terms arise. The quark
fields the following standard rotation are,

$\left(\begin{array}[]{l}u\\\ c\\\
t\end{array}\right)_{L}=V_{L}^{u}\left(\begin{array}[]{c}u^{\prime}\\\
c^{\prime}\\\ t^{\prime}\end{array}\right)_{L},\left(\begin{array}[]{l}d\\\
s\\\ b\end{array}\right)_{L}=V_{L}^{d}\left(\begin{array}[]{l}d^{\prime}\\\
s^{\prime}\\\ b^{\prime}\end{array}\right)_{L},$

where $V_{L}^{u}$ and $V_{L}^{d}$ are the $3\times 3$ unitary matrices such
that for the Cabibbo-Kobayashi-Maskawa (CKM) matrix we have
$V_{\mathrm{CKM}}\equiv V_{L}^{u\dagger}V_{L}^{d}$. Note that only the left-
chiral terms of the Lagrangian (LABEL:eq:NC-Lagrangian) lead to the quark
flavor violation. The right-chiral quark couplings to $Z^{\prime}$ in Eq.
(LABEL:eq:NC-Lagrangian) are independent of flavor and therefore are flavor-
diagonal in the mass eigenstate basis. We can write these terms in the form,

$\mathcal{L}_{Z^{\prime}}\supset\frac{g}{C_{W}}(\overline{D_{L}^{\prime}}\gamma^{\mu}Y^{D}_{L}D_{L}^{\prime})Z^{\prime}_{\mu},$
(22)

with $D_{L}^{\prime}=(d^{\prime},s^{\prime},b^{\prime})^{T}$, and

$\displaystyle Y^{D}_{L}=$ $\displaystyle\frac{1}{6\sqrt{3-4S_{W}^{2}}}\times$
$\displaystyle\mathrm{Diagonal}(-3+4S_{W}^{2},-3+4S_{W}^{2},3-2S_{W}^{2}).$

Changing the basis we get,

$\mathcal{L}_{Z^{\prime}}\supset\frac{g}{C_{W}}(\overline{D_{L}}\gamma^{\mu}Y^{D\prime}_{L}D_{L})Z^{\prime}_{\mu}=\Delta^{sd}_{L}(\overline{s_{L}}\gamma^{\mu}d_{L})Z^{\prime}_{\mu}+...,$
(24)

where $D_{L}^{\prime}=V_{L}^{d}D_{L}$ and
$Y^{D\prime}_{L}=V_{L}^{d\dagger}Y^{D}_{L}V_{L}^{d}$. Using the unitarity of
the $V_{L}$ matrix, we finally find the coupling between the quarks $d$ and
$s$, which is given by,

$\Delta_{L}^{sd}=\frac{gC_{W}}{3-4S_{W}^{2}}V_{L32}^{*}V_{L31},$ (25)

Analogously for the neutrino-$Z^{\prime}$ coupling we have,

$\Delta_{L}^{\nu\bar{\nu}}=\frac{g}{2C_{W}}\frac{1-2S_{W}^{2}}{\sqrt{3-4S_{W}^{2}}}.$
(26)

In principle, we can vary the entries of the matrix $V_{L}^{d}$ freely since
the CKM matrix does not constrain $V^{d}_{L}$, but the product
$V^{d}_{L}V^{u}_{L}$. So, we choose the following parametrization for the
$V_{L}^{d}$ matrix [46]

$V_{L}^{d}=\left(\begin{array}[]{ccc}\tilde{c}_{12}\tilde{c}_{13}&\tilde{s}_{12}\tilde{c}_{23}e^{i\delta_{3}}-\tilde{c}_{12}\tilde{s}_{13}\tilde{s}_{23}e^{i(\delta_{1}-\delta_{2})}&\tilde{c}_{12}\tilde{c}_{23}\tilde{s}_{13}e^{i\delta_{1}}+\tilde{s}_{12}\tilde{s}_{23}e^{i(\delta_{2}+\delta_{3})}\\\
-\tilde{c}_{13}\tilde{s}_{12}e^{-i\delta_{3}}&\tilde{c}_{12}\tilde{c}_{23}+\tilde{s}_{12}\tilde{\tilde{s}}_{13}\tilde{s}_{23}e^{i(\delta_{1}-\delta_{2}-\delta_{3})}&-\tilde{s}_{12}\tilde{s}_{13}\tilde{c}_{23}e^{i(\delta_{1}-\delta_{3})}-\tilde{c}_{12}\tilde{s}_{23}e^{i\delta_{2}}\\\
-\tilde{s}_{13}e^{-i\delta_{1}}&-\tilde{c}_{13}\tilde{s}_{23}e^{-i\delta_{2}}&\tilde{c}_{13}\tilde{c}_{23}\end{array}\right).$
(27)

where $\tilde{s}_{ij}=\sin{\tilde{\theta}_{ij}}$,
$\tilde{c}_{ij}=\cos{\tilde{\theta}_{ij}}$ and $\delta_{i}$ are the phases,
with $i,j=1,2,3$. For our purposes, the following entries will be important,

$\displaystyle V_{L31}^{d}$ $\displaystyle=$
$\displaystyle-\tilde{s}_{13}e^{-i\delta_{1}}$ (28) $\displaystyle
V_{L32}^{d}$ $\displaystyle=$
$\displaystyle-\tilde{c}_{13}\tilde{s}_{23}e^{-i\delta_{2}}$ (29)

then, the product which appears in the $\Delta_{L}^{sd}$ coupling is

$V_{L31}^{d}V_{L32}^{d*}=-\tilde{s}_{13}\tilde{c}_{13}\tilde{s}_{23}e^{-i(\delta_{1}-\delta_{2})}\equiv|V_{L32}^{d*}V_{L31}^{d}|e^{-i\delta}$
(30)

where we leave the product $|V_{L32}^{d*}V_{L31}^{d}|$ and the phase $\delta$
as free parameters.

##  III Kaon decays

The rare Kaon decay modes $K^{+}\to\pi^{+}\nu\bar{\nu}$ and
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ are considered golden modes in flavor
physics, as they are very well understood theoretically and are sensitive to
new physics contributions. In the SM both decays occur only at loop level and
are dominated by $Z$ penguin and box diagrams. The corresponding branching
ratios have been calculated at a high precision, including NNLO QCD,
electroweak corrections and also non-perturbative and isospin breaking effects
[47, 48, 49, 50, 51].

The decay $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ is CP conserving, whereas
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ is CP violating. In the 331 model, the
new sources of flavor and CP violation which contribute to these decays come
from the $Z^{\prime}$ interactions with ordinary quarks and leptons, as
discussed above. Although these couplings induce the transitions at tree
level, they are suppressed by the large $Z^{\prime}$ mass.

Following the notation of Ref. [52], we can write the branching ratios for the
Kaon decay modes $K\to\pi\nu\bar{\nu}$ as,

$\begin{split}&BR(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})=\\\ &\text{\ \ \ \ \ \
\ \ \ \ \ \ \
}\kappa_{+}\left[\left(\frac{\text{Im}X_{eff}}{\lambda^{5}}\right)^{2}+\left(\frac{\text{Re}X_{eff}}{\lambda^{5}}-\bar{P}_{c}(X)\right)^{2}\right],\end{split}$
(31)

and,

$BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})=\kappa_{L}\left(\frac{\text{Im}X_{eff}}{\lambda^{5}}\right)^{2}.$
(32)

In these expressions $\lambda$ denotes the Cabibbo angle, $\kappa_{+}$ and
$\kappa_{L}$ are given by,

$\kappa_{+}=(5.21\pm 0.025)\cdot
10^{-11}\left(\frac{\lambda}{0.2252}\right)^{8},$ $\kappa_{L}=(2.25\pm
0.01)\cdot 10^{-10}\left(\frac{\lambda}{0.2252}\right)^{8},$

and $P_{c}(X)$ summarizes the charm contribution,

$\bar{P}_{c}(X)=\left(1-\frac{\lambda^{2}}{2}\right)P_{c}(X),$

with,

$P_{c}(X)=(0.42\pm 0.03)\left(\frac{0.2252}{\lambda}\right)^{4}.$

$X_{\text{eff}}$ describes the contribution of short distance physics,

$X_{\text{eff}}=V_{ts}^{*}V_{td}X_{L}(K),$

where,

$X_{L}(K)=\eta_{X}X_{0}(x_{t})+\frac{\Delta_{L}^{\nu\bar{\nu}}}{g_{SM}^{2}m_{Z^{\prime}}^{2}}\frac{\Delta_{L}^{sd}}{V_{ts}^{*}V_{td}},$
(33)

and,

$g_{SM}^{2}=4\frac{m_{W}^{2}G_{F}^{2}}{2\pi^{2}}=1.78137\times 10^{-7}\text{\
GeV}^{-2}.$

The first term in Eq. (33) represents the SM contribution, which is dominated
by $Z$-penguin and box diagrams, and includes QCD and electroweak corrections.
The factor $\eta_{X}$ is close to unity, $\eta_{X}=0.994$, and

$X_{0}(x_{t})=\frac{x_{t}}{8}\left[\frac{x_{t}+2}{x_{t}-1}+\frac{3x_{t}-6}{(x_{t}-1)^{2}}\ln
x_{t}\right],$

with $x_{t}=m_{t}^{2}/m_{W}^{2}$. The 331 contribution is enclosed in the
second term of Eq. (33), with $\Delta_{L}^{sd}$ and
$\Delta_{L}^{\nu\bar{\nu}}$ given in the Eqs. (25) and (26), respectively. If
$Z^{\prime}$ is absent we have $\Delta_{L}^{\nu\bar{\nu}}=\Delta_{L}^{sd}=0$
and the SM result is recovered.

The decays $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ are related to each other in the SM, via
isospin symmetry, since both transitions are ruled by the same short distance
operator. This interdependence leads to the Grossman-Nir limit [53],

$BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})\leq
4.3BR(K^{+}\rightarrow\pi^{+}\nu\bar{\nu}).$ (34)

For a fixed value of $BR(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$, this
theoretical bound provides an upper limit for
$BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$, which is typically still stronger
than the current experimental limits. The bound remains valid in SM extensions
in which the new physics is much heavier than the Kaon mass. In particular,
the 331 model obeys the bound, as we will show in the next section.

Having presented the formulas that summarizes the predictions of the 331 model
regarding meson FCNC processes, we can now discuss the implications of the
experimental searches for the flavor violating Kaon decays on the parameter
space of the model.

Figure 2: Branching ratio versus $Z^{\prime}$ mass for kaon long (left panel),
BR($K_{L}\to\pi^{0}+\bar{\nu}\nu$), and for the $K^{+}$,
BR($K^{+}\to\pi^{+}+\bar{\nu}\nu$) (right panel). The gray band provides the
excluded region by KOTO experiment, and the blue region provides the allowed
by the NA62 experiment. The colored bar represents the variation of the
$\delta$ phase, for details see the text.

##  IV Results

In this section, we present our results for the FCNC processes using Eqs. (31)
and (32), for the 331 r.h.n model, but we emphasize that our findings are also
applicable to the 3-3-1LHN [14], which is a 3-3-1 version that features heavy
neutral leptons instead of right-handed neutrinos. Anyway, we compare our
results with the last limits from KOTO [54] and, NA62 Run1 (2016 + 2017 +
2018)222https://indico.cern.ch/event/868940/contributions/3815641/attachments/2080353/3496097/RadoslavMarchevski_ICHEP_2020.pdf
[11, 12]. In Fig. 2, we display the branching for $K_{L}$ (right panel) and
$K^{+}$ (left panel) versus the $Z^{\prime}$ mass for the 331 r.h.n model,
namely combining the standard model contribution and the new one provided by
the new $Z^{\prime}$ boson, as mentioned above. We compute both branching
varying the $Z^{\prime}$ mass, the phase $\delta$ (color bar) and we fix the
product $|V_{L32}^{d*}V_{L31}^{d}|=10^{-3}$. From the left-panel of Fig. 2 one
can conclude that the KOTO experiment excludes at most $Z^{\prime}$ masses
around $400$ GeV, which occurs for $\delta>1.4$. However, from the right-panel
of Fig. 2, we find that $Z^{\prime}$ masses below $3\times 10^{3}$ GeV might
be excluded depending on the value adopted for the phase $\delta$. For
$\delta\rightarrow\pi/2$, the NA62 sensitivity to heavy $Z^{\prime}$ mediators
severely weakens. Hence, NA62 yields complementary limits to other existing
probes [55, 56, 35]. We can also see the presence of a dip due to destructive
interference between the $Z^{\prime}$ and the SM contributions, where the
branching ratio lies bellow the SM predicted value. In the plot it occurs for
$Z^{\prime}$ masses in the $600-900$ GeV range, but in general its depth and
location depends on the combination of $|V_{L32}^{d*}V_{L31}^{d}|$ and
$\delta$. Notice that there is no dip when $\delta=\pi/2$, while it reaches
its maximum depth for $\delta=0$.

Figure 3: Excluded parameter space regions in the 331 r.h.n model, in the
plane $|V_{L32}^{d*}V_{L31}^{d}|$ versus $m_{Z^{\prime}}$, for $\delta=0$
(left panel) and $\delta=\pi/2$ (right panel). The colored regions represent
the limits from NA62 (cyan), KOTO (blue) and LHC current (gray) and prospects
(dashed lines).

We have drawn quantitative conclusions based on a particular value for
$V_{L32}^{d*}V_{L31}^{d}$. To assess how our bounds change for different
choices of this product we examine the
range$10^{-4}<V_{L32}^{d*}V_{L31}^{d}<10^{-1}$. In Fig. 3 we show exclusion
plots in the plane $|V_{L32}^{d*}V_{L31}^{d}|$ versus $m_{Z^{\prime}}$, for
fixed $\delta=0$ (left-panel) and $\delta=\pi/2$ (right-panel). Here we
combined limits from NA62, KOTO, and LHC, High-Luminosity LHC (HL-LHC) and
High-Energy-LHC (HE-LHC). The strongest collider bounds on the $Z^{\prime}$
mass stems from the resonant production of $Z^{\prime}$ decaying into
dileptons [40].

There is an interesting interplay between collider and flavor physics. While
collider bounds rely mostly on the interaction strength between the
$Z^{\prime}$ boson and fermions, which is fixed in 3-3-1 models, the flavor
bounds weaken with $|V_{L32}^{d*}V_{L31}^{d}|$. The collider bounds displayed
in Fig. 3 are conservative, as they take into account the presence of
$Z^{\prime}$ decays into right-handed neutrinos and exotic quarks, which can
be light enough so that the decays are not kinematically forbidden. The
original lower mass bound reads $m_{Z^{\prime}}>4$ TeV [55], but those exotic
decays were ignored in [55]. If more decay channels become available, then the
bound weakens. Here, in order to take into account this uncertainty on the
$Z^{\prime}$ decay modes, we assume conservatively a branching fraction of 50%
into charged leptons, which leads to the grey region in Fig. 3 in agreement
with [41]. We also show the prospects for the HL-LHC, with $3$ ab-1 of
integrated luminosity, and the HE-LHC, which corresponds to an integrated
luminosity of $15$ ab-1 at a center-of-mass energy of $27$ TeV. These
projected collider limits were obtained using the code described in [57],
which can be used to forecast lower mass limits for resonance searches, which
is precisely our case.

We found that the NA62 bounds from the decay
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ are rather restrictive, providing
stronger limits than the ones from colliders over a significant region of the
parameter space. Looking at left-panel of Fig.3, for instance, where
$\delta=0$, NA62 can exclude $Z^{\prime}$ masses as high as $10$ TeV, if
$|V_{L32}^{d*}V_{L31}^{d}|$ is of the order $10^{-2}$ or larger. In the same
vein, NA62 enforces the product $|V_{L32}^{d*}V_{L31}^{d}|$ to remain below
$\sim 10^{-3}$, when $m_{Z^{\prime}}\sim 2$ TeV.

On the other hand, these parameters are completely unconstrained by KOTO when
$\delta=0$, since the contribution from $Z^{\prime}$ to the decay
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ vanishes in this case. In the absence of
new CP violating sources, the $BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$ takes
the same value as in the SM, since this process will occur only through the
contribution to CP violation from the SM via CKM matrix.

This can be easily understood with Eq. (32), from which we see that
$BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$ depends only on the imaginary part
of $X_{eff}$, in particular, on $Im(V_{ts}^{*}V_{td})$ when $\delta=0$. Still
in the left-panel of Fig. 3 we notice that in the range shown, the Grossman-
Nir limit does not appear, because the suppressed values of
$BR(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$ makes this bound easily satisfied
for practically any reasonable value of $|V_{L32}^{d*}V_{L31}^{d}|$. However,
as $\delta$ increases from $0$ to $\pi/2$, this bound becomes more relevant,
likewise the KOTO exclusion region gradually grows, reflecting the enhancement
in the $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ decay, while the NA62 exclusion
region slightly decreases. Nevertheless, even with maximum enhancement at
$\delta=\pi/2$ (right plot), the KOTO bounds remain less constraining compared
to NA62 and the Grossman-Nir limit.

##  V Discussion

Our conclusions relied on the presence of flavor changing interactions
involving the $Z^{\prime}$ boson, but as the model features a large scalar
sector, that could be new sources of flavor changing interactions rising from
the heavy scalar fields. These new contributions have been shown to be
subdominant. Thus can be safely ignored. Moreover, the entire reasoning was
based on the 3-3-1 model with right-handed neutrinos, but our results are also
applicable to the 3-3-1 model where the right-handed neutrinos are replaced by
heavy neutral fields. This occurs because these models have the same neutral
current. In summary, our finding are relevant for two different models and
solid irrespective of the presence of scalar fields.

##  VI Conclusion

In this work we explored the FCNC processes mediated by a $Z^{\prime}$ gauge
boson featuring both in the 3-3-1 r.h.n and 3-3-1LHN models. We computed the
$K^{+}$ and $K_{L}$ decay rates to missing energy, considering the extra
contributions from the $Z^{\prime}$ in addition to the SM contribution,
leaving the quark mixing matrix and the $Z^{\prime}$ mass as free parameters.
We performed a complementary analysis using the results from NA62, KOTO, and
the LHC (current and prospects) to set bounds on the 331 r.h.n parameters. We
found that the last result of the NA62 experiment was able to constrain a
large region of the parameter space, setting lower limits on the $Z^{\prime}$
mass that can be more stringent than those from dilepton searches at the LHC.
For example, we can impose $m_{Z^{\prime}}>10$ TeV for
$|V_{L32}^{*}V_{L31}|\sim 10^{-1}$, while $|V_{L32}^{*}V_{L31}|\lesssim
2\times 10^{-3}$ for $m_{Z^{\prime}}=3$ TeV. These results apply for
$\delta=0$, where the sensitivity of NA62 is maximum. In the case when the new
CP violation effects are large our constraints weaken.

###### Acknowledgements.

The authors thank Antonio Santos and Diego Cogollo for discussions. TM, CS and
FSQ thanks UFRN and MEC for the financial support. FSQ also acknowledges the
CNPq grants 303817/2018-6 and 421952/2018-0, the financial support from ICTP-
SAIFR FAPESP grant 2016/01343-7, and the Serrapilheira Institute (grant number
Serra-1912-31613). FSQ and CS have been supported by the São Paulo Research
Foundation (FAPESP) through Grant No 2015/15897-1. CS is supported by grant
2020/00320-9, São Paulo Research Foundation (FAPESP). SK acknowledges the
support of the FONDECYT (Chile) grant No 1190845. Y. S. Villamizar
acknowledges the ﬁnancial support from CAPES under grants
88882.375870/2019-01. This work was supported by the Serrapilheira Institute
(grant number Serra-1912-31613). We thank the High Performance Computing
Center (NPAD) at UFRN for providing computational resources.

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