sure the clients are disjoint across rounds. Note that DP-FTRL would run less
than one epoch for StackOverflow. On each client, the number of local epochs
is fixed to be one and the batch size is sixteen, and we constrained the
maximum number of samples on each client to be 256. The momentum for both DP-
SGDM and DP-FTRLM is fixed to 0.9.

In most of the experiments, we will tune server learning rate, client learning
rate and clip norm for a certain noise multiplier. We tune a relative large
grid (client learning rate in $\\{0.1,0.2,0.5,1,2\\}$, server learning rate in
$\\{0.03,0.1,0.3,1,3\\}$, clip norm in $\\{0.1,0.3,1,3,10\\}$) when the noise
multiplier is zero. And we have several observation: the best accuracy of clip
norm 0.3 and 1.0 are slightly better than larger clip norms, which suggests
that clip norm could generally help for this language task; increasing server
learning rate could complement decreasing clip norm when clip norm is
effective; the largest client learning rate that does not diverge often leads
to good final accuracy. As adding noise increases the variance of gradients,
we often have to decrease learning rate in practice. Based on this heuristic
and the observation from tuning when noise multiplier is zero, we choose
client learning rate from $\\{0.1,0.2,0.5\\}$, server learning rate from
$\\{0.1,0.3,1,3\\}$ and clip norm from $\\{0.3,1,3\\}$ unless otherwise
specified. We use DP-SGD with zero noise for StackOverflow, as gradient
clipping can improves accuracy for language tasks.

###  F.2 Centralized Training with Large Number of Epochs by Interleaving
Restarting and Non-restarting

Appendix D.3 describes the idea of interleaving between restarting and non-
restarting. Here, we examine how such schedules might affect the model
utility. Additionally, we consider the effect of the tree completion trick
(Section D.3.1). We consider CIFAR-10 and EMNIST, which are hard datasets that
might require a large number of epochs to learn. For CIFAR-10, we fix the
batch size to be $500$, number of epochs to be $100$ (thus $10000$ steps in
total); for EMNIST, we fix the batch size to be $500$ and number of epochs to
be $50$ (thus $69750$ steps in total). On each dataset, similar as in Section
7, we compare DP-SGD with or without amplification, and DP-FTRL(M) with
different restarting schedules.

(a) CIFAR-10, $100$ epochs.

(b) EMNIST, $50$ epochs.

Figure 9: Interleaving between restarting and non-restarting. r1, r5, r20, and
r25 corresponds to restarting every one, five, twenty, and twenty five epochs
respectively. non-r corresponds to the version of DP-FTRL with no restarting.
completion refers to the “completion trick” to the closest power of two(from
Section D.3.1).

In Figure 9(a), we plot the results for DP-FTRL(M) with non-restarting and
restarting every $1$, $5$, $20$ epochs (solid lines), and comparing them with
DP-SGD with and without amplification. Additionally, we use the tree
completion trick for restarting every $5$ and $20$ epochs (dashed lines). We
can see the following.

  * •

Neither non-restarting nor restarting every epoch yields accuracy that are
comparable to DP-SGD. On the other hand, without the tree completion trick,
restarting every $20$ epochs gives much better accuracy, which means that
interleaving between restart and non-restart is crucial.

  * •

The tree completion trick helps for restarting every $5$ epochs and hurts for
restarting every $20$ epochs, demonstrating the trade-off we mentioned before.

  * •

Overall, without the tree completion trick, restarting every $5$ epochs with
the tree completion trick gives the best accuracy, and we can see a “cross-
over” between it and DP-SGD similar as that in Figure 1, yet at a larger
$\varepsilon\approx 18$. With the completion trick, restarting every $20$
epochs gives the best accuracy. A “cross-over” happens at $\varepsilon\approx
14$.

  * •

Similar as in Figure 1, we can see that DP-FTRL is always better than DP-SGD
without amplification.

In Figure 9(b), we plot the results on EMNIST for restarting every $1$, $5$,
$25$ epochs and non-restarting. We can observe similar trend as in the
CIFAR-10 experiments. Namely, the tree completion tricks helps in some cases,
and the best accuracy is achieved by restarting every $5$ epochs with the tree
completion trick, which outperforms DP-SGD with amplification starting from
$\varepsilon\approx 5$.

###  F.3 Omitted Details for StackOverflow Experiments

Figure 10: Test and Validation accuracy for the StackOverflow next word prediction task under different privacy epsilon by varying noise multipliers. Server Optimizer | Epsilon | Accuracy | Hyperparameters
---|---|---|---
Validation | Test | Noise | ServerLR | ClientLR | Clip
DP-SGDM | 18.20 | 17.52 | 18.89 | $0.3$ | $1$ | $0.5$ | $0.3$
DP-FTRLM | 18.71 | 20.10 | 21.33 | $1.13$ | $0.3$ | $0.5$ | $1$
DP-SGDM | 7.51 | 16.94 | 18.30 | $0.4$ | $0.1$ | $0.5$ | $1$
DP-FTRLM | 7.83 | 19.01 | 20.22 | $2.33$ | $1$ | $0.5$ | $0.3$
DP-SGDM | 3.93 | 16.39 | 17.94 | $0.5$ | $0.3$ | $0.5$ | $0.3$
DP-FTRLM | 4.19 | 17.34 | 18.49 | $4.03$ | $0.1$ | $0.5$ | $1$
DP-SGDM | 2.40 | 16.08 | 17.48 | $0.6$ | $0.3$ | $0.5$ | $0.3$
DP-FTRLM | 2.60 | 16.45 | 17.60 | $6.21$ | $0.1$ | $0.5$ | $1$
DP-SGDM | 1.61 | 15.78 | 17.13 | $0.7$ | $0.3$ | $0.5$ | $0.3$
DP-FTRLM | 1.77 | 15.43 | 16.52 | $8.83$ | $0.3$ | $0.5$ | $0.3$
Table 4: Validation and test accuracy for the StackOverflow next word
prediction task under different privacy epsilon.

We compare the accuracy of the momentum variant of DP-FTRL with the momentum
variant of DP-SGD as baseline under different privacy epsilon. We tune
hyperparameters as described in Section F.1 and select the hyperparameters
achieve the best validation accuracy for StackOverflow (see Table 4 and Figure
10). DP-FTRLM performs better than DP-SGDM when the epsilon is relatively
large, but performs worse when the epsilon is small ($\varepsilon<2.60$ in
Table 4). More noise are added to DP-FTRLM to achieve the same privacy epsilon
as DP-SGDM. However, DP-FTRLM can result in utility (accuracy) not (much)
worse than DP-SGDM without relying on amplification by sampling, which makes
it appealing for practical federated learning setting where population and
sampling is difficult to estimate [5]. Note that the noise added for both DP-
FTRLM and DP-SGDM are considered large for federated learning tasks. The
effective noise could be significantly reduced by sampling more clients each
round in practice [51], and more discussion on this front is in Appendix G.

##  Appendix G Omitted Details for Experiments in Section 7.3

###  G.1 Details of Hyperparameter Tuning

In Section F.3, a significant amount of noise has to be added in both DP-FTRLM
and DP-SGDM to achieve nontrivial privacy epsilons, which leads to undesired
accuracy degradation. For example, the test accuracy of DP-FTRLM on
StackOverflow dataset decreases from $25.15\%$ when $\varepsilon=\infty$ to
$20.22\%$ when $\varepsilon=8.5$ when the number of clients per round is fixed
at 100. In practical federated learning tasks, the total population is very
large and many more clients could be sampled every round. In this section,
taking StackOverflow as an example, we study the minimum number of sampled
clients per round (report goal in [10]) to achieve a target accuracy under
certain privacy budget.

(a) SGDM with zero noise

(b) FTRLM with zero noise

Figure 11: Training curve of the best validation accuracy under various clip
norm for StackOverflow.

##### Fix the clip norm and client learning rate to reduce hyperparameter
tuning complexity.

We first find the largest noise multiplier that would meet the target accuracy
based on selecting 100 clients per round. As an extensive grid search over
noise multiplier while simultaneously tuning server learning rate, client
learning rate and clip norm is computationally intensive, we fix the clip norm
to 1 and the client learning rate to 0.5 based on Figure 11. We then tune the
server learning rate from $\\{0.3,1,3\\}$ for each noise multiplier.

##### Grid search for the largest noise multiplier to meet the target.

We use a grid of ten noise multipliers between $0$ ($\varepsilon=\infty$, test
accuracy=$24.89$) and $0.3$ ($\varepsilon=18.89$, test accuracy=$18.89$) for
DP-SGDM, and between $0$ ($\varepsilon=\infty$, test accuracy=$25.15$) and
$1.13$ ($\varepsilon=19.74$, test accuracy=$21.33$) for DP-FTRLM. And we
further add five noise multipliers between $0$ and $0.035$ for DP-SGDM, and
between $0$ and $0.149$ for DP-FTRLM based on the results of the previous grid
search on ten noise multipliers. The test accuracy is presented in Figure
2(b). We set the target test accuracy as $24.5\%$ and select noise multiplier
$0.007$ (with server learning rate $3$) for DP-SGDM and noise multiplier
$0.149$ (with server learning rate $3$) for DP-FTRLM.

##### Report goal for the nontrivial privacy epsilon in practice.

The standard deviation of noise added in each round is proportional to the
inverse of the number of clients per round (report goal). The practical
federated learning tasks often have a very large population and report goal,
and we could simultaneously increase the noise multiplier and report goal, so
that the utility (accuracy for classification and prediction tasks) will
likely not degrade [51] while the privacy guarantee is improved. The
validation accuracy of simulation performance with two different report goals
for StackOverflow is presented in Figure 12. The noise multiplier 0.149 is
used for DP-FTRLM and 0.007 is used for DP-SGD when report goal is 100, which
is the largest noise multiplier to meet the target test accuracy determined by
Figure 2(b). We run each experiment for five times and plot the curves for the
median validation accuracy, the corresponding test accuracy are $24.73\%$ for
DP-SGDM and $24.51\%$ for DP-FTRLM. We then run the same experiments with
report goal of 1000, and proportionally increase the corresponding noise
multiplier to be 1.49 for DP-FTRLM and 0.07 for DP-SGDM. The performance of
1000 report goal is slightly better with test accuracy $25.19\%$ for DP-SGDM
and $24.67\%$ for DP-FTRLM. We will assume the utility will not decrease if
report goal and noise multiplier are simultaneously and proportionally
increased.

As shown in Table 5, both report goals 100 and 1000 would provide trivial
privacy guarantee of large epsilon for the target utility. We have to increase
the report goal to $2.06e4$ to get a nontrivial privacy epsilon (less than 10)
with DP-FTRLM and the StackOverflow population of $3.42e5$ 101010The best
epsilon DP-SGDM can achieve is $10.16$ by increasing report goal to be as
large as the population $3.42e5$. Smaller report goal could achieve similar
privacy guarantee if the population becomes larger. In Figure 2(c), the
relationship between privacy guarantee and report goal for DP-FTRLM and DP-
SGDM are presented. DP-FTRLM provides better privacy guarantee by smaller
report goal when the privacy epsilon is relatively large or very small. The
range where DP-FTRLM outperforms DP-SGDM in report goals and privacy
guarantees are larger when the population is relatively small or very large.

Figure 12: Training curves of validation accuracy for DP-SGDM and DP-FTRLM for
StackOverflow for report goal 100 and 1000 (suffix C100 and C1000 in the
legend). DP-FTRLM with restart (see Appendix D) is used when report goal is
1000 (less than five epochs of data). Simultaneously increasing noise
multiplier and report goal by 10x could significantly improve the privacy
guarantee without sacrificing the utility. The noise multiplier for DP-
SGDM-C100, DP-FTRLM-C100, DP-SGDM-C1000, DP-FTRLM-C1000 are 0.007, 0.149,
0.07, and 1.49, respectively. The corresponding test accuracy are 24.73%,
24.51%, 25.19% and 24.67%. The corresponding privacy $\varepsilon$ can be
found in Table 5

.

Server Optimizer | Privacy | Setting
---|---|---
Epsilon | Delta | Noise | Report goal | Population
DP-SGDM | 1.78e7 | 1e-6 | 0.007 | 100 | 3.42e5
DP-FTRLM | 363.66 | 1e-6 | 0.149 | 100 | 3.42e5
DP-SGDM | 7.71e4 | 1e-6 | 0.07 | 1000 | 3.42e5
DP-FTRLM | 32.52 | 1e-6 | 1.49 | 1000 | 3.42e5
DP-SGDM | 9.42 | 1e-6 | 23.97 | 3.42e5 | 3.42e5
DP-FTRLM | 9.76 | 1e-6 | 7.53 | 5.06e3 | 3.42e5
DP-FTRLM | 4.11 | 1e-6 | 24.29 | 1.63e4 | 3.42e5
DP-SGDM | 8.11 | 1e-6 | 0.67 | 9.56e3 | 1e6
DP-FTRLM | 7.57 | 1e-6 | 5.73 | 3.85e3 | 1e6
DP-SGDM | 3.70 | 1e-6 | 1.20 | 1.71e4 | 1e6
DP-FTRLM | 3.56 | 1e-6 | 15.29 | 1.03e4 | 1e6
Table 5: The ($\varepsilon,\delta$) privacy guarantee for DP-FTRLM and DP-SGDM
under realistic and hypothetical report goal and population of StackOverflow
that would meet the target test accuracy 24.5%. Note that the DP-FTRLM privacy
accounting is based on the restart strategy in Appendix D.


11institutetext:  Faculty of Mathematics, Physics and Informatics
Comenius University in Bratislava, Slovakia
11email<EMAIL_ADDRESS>
http://cogsci.fmph.uniba.sk/cnc/

# QuasiNet: a neural network with
trainable product layers

Kristína Malinovská  0000-0001-7638-028X Slavomir Holenda
L’udovít Malinovský  0009-0000-1911-8453

###### Abstract

Classical neural networks achieve only limited convergence in hard problems
such as XOR or parity when the number of hidden neurons is small. With the
motivation to improve the success rate of neural networks in these problems,
we propose a new neural network model inspired by existing neural network
models with so called product neurons and a learning rule derived from
classical error backpropagation, which elegantly solves the problem of
mutually exclusive situations. Unlike existing product neurons, which have
weights that are preset and not adaptable, our product layers of neurons also
do learn. We tested the model and compared its success rate to a classical
multilayer perceptron in the aforementioned problems as well as in other hard
problems such as the two spirals. Our results indicate that our model is
clearly more successful than the classical MLP and has the potential to be
used in many tasks and applications.

###### Keywords:

product neuron multiplication error backpropagation XOR parity 2 spirals.

##  1 Introduction

Multiplication of neuronal activations has been theoretically studied, but it
is used relatively rarely in practice. In the simplest case, a neuron similar
to a classical perceptron computes a weighted sum of the terms of a higher-
order polynomial, and only the summation weights are trained via error
backpropagation. Since the number of possible terms grows exponentially with
increasing polynomial degree, the composition of the polynomial can be
constrained in various ways. In the more complex case of the so-called product
neurons, the network would learn the exponents of the individual components of
each term. However, this leads to computations in the domain of complex
numbers in the general case if a negative number is raised to a fraction. As a
result, multiplication is only used in particular cases as an optimization
technique at pre-selected exact locations in the model. Most of the time, the
network itself does not learn to use multiplication. Despite these obstacles,
both theoretical and practical results argue in favor of using multiplication
as a tool to increase the computational power of neural networks.

In this paper, we propose a novel neural network model QuasiNet that has
layers of product neurons as well as layers of classical summation neurons.
The novelty of our approach is that we define a new way of applying weights at
the product layer, which we call quasi-exponentiation. Unlike other models,
our method allows trainable product neuron weights using a gradient descent
method without the need for computations in the domain of complex numbers.
Thus, the network is able to partially learn the polynomial composition, which
increases the generality of the model. The multiplication layer is designed in
such a way that it can be plugged into the architecture as arbitrarily and
easily as a classical Sigmoidal summation layer. The results of our
experiments show the incomparably higher efficiency of our model in XOR and
parity problems compared to a classical two-layer perceptron with the same
architecture and parameters, as well as in other hard problems such as the
separation of two-spirals.

##  2 Related work

Multiplication has been used in neural models since the late 1950s, and in the
1960s it was popular to use higher-order Sigmoidal neurons for pattern
recognition [13]. These were classical Sigmoidal neurons, but the input vector
was augmented with terms generated by multiplying and exponentiating the
inputs, and these terms were given their own weights. This approach is a
natural extension of the classical perceptron, but in the general case leads
to an exponential increase in dimensionality, with the upper bound on the
degree of the polynomial being mainly a designer’s choice. In such networks,
back propagation of error can be easily applied, and neurons of this type are
also known as _sigma-pi_ neurons [16], since they perform a weighted sum of
the products of the inputs.

The solution to the dimensionality problem was a family of models in which the
number of polynomial terms was restricted to some pre-selected ones. For
example, Giles et al. [9] and Spirkovska and Reid [21] show that classical
perceptron networks can be significantly outperformed in both learning speed
and generalization capability in this way. However, this approach requires
some knowledge of the problem to be embedded in the model by its designer.
Constructive models have been a response to this. Their essence was to learn
in different ways which polynomial terms were needed, starting with a minimal
model and gradually adding higher order terms and evaluating their
effectiveness [15, 10].

A generalization of the above principle is the product neuron model [6], which
assumes trainable real weights acting as exponents of the inputs. The authors
have solved the problem with the input to the domain of complex numbers by
using only the real part of the number. For logical inputs, such a neuron
becomes a de facto summation neuron which has cosine as an activation
function, leading to further theoretical problems [2].

An alternative approach was suggested by Ghosh and colleagues [8]. Their model
first linearly combines the inputs and then directly multiplies this linear
combination without using an activation function. The nonlinear activation
function is applied only at the output. The only primes of the model are the
linear weights, and the weights on the multiplication layer are fixed to one.
The result can be analyzed as a polynomial. Hence the product of the sums
stands out in the formula, it is said to be a pi-sigma network. Since the
weights on the multiplication layer are fixed and only linear weights are
learned, the learning algorithm is based on the gradient method.

The authors tested the success of the network on many common tasks such as
feature approximation, data classification, feature recognition with
translation and rotation invariance, and also verified the network’s ability
to learn logical functions of parity, symmetry and negation. The achieved
success rate is an order of magnitude better in the case of parity compared to
the classical multilayer perceptron and comparable to other models with higher
order neurons.

From our point of view, the drawback is that [8] fixes the multiplicative
weights and does not even try to find a way to learn them. Moreover, the
absence of an activation function after the linear transformation reduces the
pi-sigma network to a polynomial model with forward-selected terms. [20]
propose a solution with a constructive model, similar to the constructive pi-
sigma models above.

Analogs of multiplication occur in biological neural networks in various
forms. One possibility are the so called dendritic clusters [12] found in the
dendritic trees, which can also divide, or could represent an operation
similar to the sigma-pi neurons [11]. The ability of neurons to multiply
inputs has been found in the auditory [22] and visual [1] cortices. Some
authors argue that the physical proximity of synapses leads to multiplicative
behavior, whereas distant synapses function in a summation mode [3].

Schmitt [19] analyzes different types of neural networks with multiplicative
and sigmoidal neurons and derives general mathematical constraints for each
type of network in terms of the Vapnik-Chervonenkis dimension. It concludes
that multiplication in neural networks is a suitable choice for increasing the
degree of nonlinear interaction and computational power, even for high-order
neurons. The paper also provides a qualitative overview of neural models with
multiplication, which is still quite relevant despite the current boom of deep
networks with a large number of parameters and the use of brute force graphics
cards.

In the context of deep neural networks, sum-product multiplication has been
studied in sum-product networks [14, 4]. Some new models use multiplication of
vectors and activation matrices ḅy elements at selected locations of the model
[7, 23, 5, 18]. Articles of varying quality can be found on the blogosphere
that describe specific applications of multiplication in deep neural networks
and show its benefits in concrete examples. Thus, this is still an active and
promising research direction.

##  3 QuasiNet

We propose a novel neural network model we call QuasiNet. Our model resembles
the pi-sigma network [8], except that we apply the hyperbolic tangent as an
activation function on the hidden layer, thus obtaining hidden activations
from the interval $(-1,1)$. We then multiply these on the output layer. The
weights in our model are not fixed, nor do they act as exponents, but are
involved in the computation so that at one extreme a given hidden layer
activation does not matter (it is reduced to one as if the neuron activations
were raised to 0) and at the other extreme it performs at its original
function (unchanged as if they were raised to 1). This recalculation, which
replaces the exponentiation from the previous models, is in fact polynomial.
This property guarantees that complex numbers do not appear anywhere in the
computation, and also allows the model to learn which hidden neurons ,,make
sense” to use to achieve the desired output. We show a schematic
representation of the model in Fig. 1. For the ease of explaining the workings
of the model we will describe a simple case with one summation and one product
layer. However, QuasiNet can be extended to any number of connected layers and
of various types (both product and summation) and still can be trained with
the same standard error backpropagation learning rule [17].

Figure 1: The schematic depiction of our model adapted from [8].

In our current research, we chose the hyperbolic tangent because by
representing the logical values truth and false by $-1$ and $+1$ we can
directly simulate the logical XOR function using multiplication, since sign
multiplication behaves exactly like XOR. For other tasks, it may be
appropriate to choose a different activation function.

###  3.1 Forward pass

Given an input $\boldsymbol{x}$, including the trainable bias, and weights
$w^{\rm hid}$ and $w^{\rm out}$ we compute the activation of neurons on the
connected hidden layer $h$ as:

$h_{i}=\tanh(\sum_{j}w^{\rm hid}_{ij}\cdot x_{j}),$ (1)

and from that the activation of the connected product layer $y$, which in this
case, is the output layer of the model, can be expressed as

$y_{i}=\prod_{j}\left(1-\sigma(w^{\rm out}_{ij})(1-h_{j})\right),$ (2)

or

$y_{i}=\prod_{j}f(h_{j},\sigma(w^{\rm out}_{ij})),$ (3)

where $f$ is defined as:

$f(h_{j},d)=1-d(1-h_{j})$ (4)

and $\sigma$ is the logistic function.

Thus, our model uses the analogy of the product neuron [6] at the output
layer, but replaces the input-weight exponentiation by a polynomial function
$f$ with one nonlinear parameter:

$f(h,\sigma(w))=1-\sigma(w)(1-h)$ (5)

This approach preserves some properties of the exponentiation, namely
$h^{1}=f(h,1)=h$ and $h^{0}=f(h,0)=1$ but does not preserve the property
$0^{d}=0$. Instead, in the case of zero input, we get $f(0,d)=1-d$.

Moreover, our method of applying weights using the function $f$ is continuous
and continuously differentiable for all zero input values, while $0^{0}$ is
undefined and hence a point of discontinuity and without derivative. By
applying the logistic function to the weight, we achieve that the weight can
take arbitrary real values, but which correspond to the exponents of $0$ and
$1$ at the extrema. This allows the output weights to learn safely using
gradient methods which would otherwise put them outside of the desired values.

###  3.2 Error backpropagation

For the output layer, we can use the gradient descent method for the mean
squared error to derive the following rule for adapting the hidden-output
weights:

$\displaystyle\frac{\partial E}{\partial w^{\rm out}_{ij}}={}$
$\displaystyle(d_{i}-y_{i})\left(\prod_{k\neq j}1-\sigma(w^{\rm
out}_{ik})(1-h_{k})\right)$ (6) $\displaystyle(h_{j}-1)\sigma(w^{\rm
out}_{ij})(1-\sigma(w^{\rm out}_{ij}))$

and back-propagating the error to the hidden layer:

$\frac{\partial E}{\partial h_{i}}=\sum_{k}(d_{k}-y_{k})\left(\prod_{l\neq
i}1-\sigma(w^{\rm out}_{kl})(1-h_{l})\right)\sigma(w^{\rm out}_{ki})$ (7)

To speed up the computation, we can replace the partial multiplication in the
learning rule by division, in case division by zero does not occur in the
computation:

$\prod_{k\neq j}1-\sigma(w^{\rm
out}_{ik})(1-h_{l})=\frac{y_{i}}{1-\sigma(w^{\rm out}_{ij})(1-h_{j})}$ (8)

Then we used the back-propagated error to adapt the input-hidden weights as:

$\Delta w^{\rm hid}_{ij}\sim\frac{\partial E}{\partial w^{\rm
hid}_{ij}}=\frac{\partial E}{\partial h_{i}}(1-h_{i}^{2})x_{j}$ (9)

Our QuasiNet model does not need to be further constrained or stabilized; it
can be approached in all respects as a multi-layered perceptron. The
multiplication layer can also be incorporated into the network in other ways,
such as alternating the summation and multiplication layers, multiplying first
and then adding, and so on. It is also possible to modify or rescale the $f$
function according to the properties we want to achieve and the model will
look very similar.

##  4 Experimental results

###  4.1 XOR and Parity

For the baseline comparison, we used a basic multilayer perceptron (MLP)
without any regularization. We implemented our model in Python programming
language111urlhttps://github.com/kik-re/prodnet by modifying the original MLP
to make the comparison as faithful as possible. Here we present mainly the
results from experiments with the XOR and $n$-parity problem and preliminary
results from other problems. In general, XOR is indeed a parity problem, but
we keep the naming for convenience.

The parity problem, i.e., determining whether a binary number has an even or
odd number of units, is a classical, rather hard problem for neural networks
and is used to test and validate new models. In our experiments, we evaluate
the convergence, i.e., how many networks out of 100 reach a stable solution.
We consider that the network converged to the solution if it gives the correct
output for ten consecutive epochs. We gave networks at most 10 thousand
training epochs, in the case of larger $n$-parity. For both types of networks,
we use a uniform learning rate $\alpha=0.9$ and a Gaussian initialization of
the weights with distribution $\mathcal{N}(0;1.0)$. These hyperparameters were
chosen based on previous experimentation.

Figure 2: Results from XOR experiments with varying hidden layer size (max.
500 epochs) in terms of convergence (left) and average training epochs to
convergence (right), including non-converging runs in the mean.

In Fig. 2, we show the effect of hidden layer size on the convergence of
networks in the XOR task. To emphasize the point that QuasiNet is not
successful solely because of encoding the labels of the problem as $-1,1$
instead of $0,1$ we include the results for the baseline with two consecutive
hyperbolic tangent layers. For the so-called minimal XOR with 2 hidden
neurons, we observe a clear superiority of our model. So far, there is no
three-layer network model known to us without residual weights that learns
this problem to 100% in the minimal architecture with 2 hidden neurons, i.e.,
that all trained networks find a solution.

In Table 1 we show the results of initial experiments for parity $2-7$, where
parity 2 represents XOR. For MLP, we no longer found an optimal hidden layer
size for parity greater than 5. Given the constraint on the number of training
epochs we set above, MLP could not learn the task.

| QuasiNet | MLP baseline
---|---|---
$n$-parity | h | convergence | h | convergence
2 | 2 | 100 | 4 | 100
3 | 4 | 100 | 9 | 100
4 | 6 | 100 | 12 | 91
5 | 7 | 100 | 50 | 44
6 | 12 | 100 | 45 | 69
7 | 15 | 100 | 45 | 33
Table 1: Results from n-parity experiments: minimum size of the hidden layer h
for maximum number of converging networks. For MLP baseline we report results
we have achieved given by the computational limits, very large hidden layer
size would lead to a slightly better performance, but not full convergence.

In figure 3 we display the results of hidden layer size experiments for parity
7 for both our model and the MLP baseline. Note, that we allowed the MLP
baseline to train for 10 thousand epochs, which is significantly more then in
the case of our QuasiNet, which in the best case converges in less then 100
epochs. Similarly, the tested hidden layer sizes for the MLP baseline and its
success differ dramatically from our model.

Figure 3: Results from parity 7 experiments: convergence (top), training
epochs (bottom) for MLP baseline (left) and QuasiNet (blue).

In figure 4 we display a more detailed experiment with QuasiNet: convergence
in parity 8 as a function of the hidden layer size. We also show the mean
training epochs. Note, that for deeper evaluation we allowed the model to
train for 5000 instead of 500 epochs. This experiment illustrates that larger
hidden layer size does not lead to better performance. The optimal hidden
layer size for out model seems to be rather close to the number of input
features.

Figure 4: Results from parity 8 experiments with varying hidden layer size
(100 nets, max. 5000 epochs) in terms of convergence (left) and average
training epochs to convergence (right), including non-converging runs in the
mean.

Last, but not least, we present overal results for parity problem with in
terms of best hidden layer size found so far for $2\leq n\leq 13$ in Fig 5. In
our experiments we observe that most of the nets converge in relative small
number of epochs even for large $n$-parities.

Figure 5: Results from $n$-parity experiments: minimum size of the hidden
layer h for maximum number of converging networks.

###  4.2 Multiple layers and 2 spirals

The famous 2 spiral problem is commonly used for testing new neural network
models. Similarly to parity it poses a problem with mutually exclusive
situations, i.e the point belongs to one spiral or to another. We display the
2 spirals dataset with 2 thousand points we have used and its distribution
into training and testing data in Fig. 6. Note, that the spiral coordinates
are transformed to the interval $(-1,1)$ similarly to inputs in parity
problem. In our preliminary experiments we have observed that this problem
requires more hidden layers for a satisfying performance level.

Figure 6: The 2 spirals dataset (2000 pts) split into 80% training and 20%
testing data.

In Fig 7 we display training progress of QuasiNet with four hidden layers of
neurons, where the hyperbolic tangent and product layers are combined one
after another. Namely, the architecture used was: 2 input neurons, 10 tanh
neurons, 80 product neurons, 5 tanh neurons and finally 1 output product
neuron. Other hyperparameters used were learning rate $\alpha=0.01$ and a
Gaussian initialization of the weights with distribution $\mathcal{N}(0;0.5)$.
The networks were trained for 10 thousand epochs. The mean accuracy for the
testing data set was 98.275% and 4 out of 10 networks achieved full
convergence, which we deem very successful.

Figure 7: Results from training of the 2 spirals problem: mean and standard
deviation over 10 nets trained for 10000 epochs, only every 10th epoch is
shown.

##  5 Conclusion and future work

Our innovative neural network model QuasiNet compensates for particular
shortcomings of existing neural networks with product neurons and proposes a
new way to train the weights of any multiplicative components in neural
networks, which avoids the problem with computations in the domain of complex
numbers. Our model is mathematically simple, easy to implement, and can be
combined with existing models, with the promise of increasing their efficiency
in tasks with mutually exclusive situations, which are in general quite
difficult for classical neural networks to solve. QuasiNet outperforms the
classical MLP baseline in the XOR and $n$-parity tasks and excels also in the
2 spiral problem task, which is considered one of the hardest problems for
neural networks in general. A major advantage of QuasiNet is that it needs a
quite small amount of parameters and training epochs to learn hard tasks. From
the preliminary queries in the literature and results from related models we
assume QuasiNet is at least comparable with the existing baselines. Yet a
thorough experimentation and comparison outside of the scope of this debut
research paper is to be done to strengthen our claims.

There is a vital need to continue the experimentation with this intriguing new
model in terms of testing of various classical neural networks problems and
comparing it to existing work. We believe that in the future our model could
be also beneficial in the domain of deep learning where it could be used
instead of classical fully connected layers in combination with convolutional
layers for feature extraction. The fact that the model requires only a
fraction of weights compared to the classical fully-connected MLP with Sigmoid
or similar activation function could be utilized for analyzing the properties
encoded in the feature space of deep networks thus contributing to better
explainability of the model. The properties of QuasiNet, its explainability
and possibility to encode mutually exclusive properties would also benefit the
neural modeling in the domain of cognitive robotics and human-robot
interaction where the development of the understanding and interpretation of
the sensory input is vital for the robotic system.

Acknowledgements The research leading to these results has received funding
from the project titled TERAIS in the framework of the program Horizon-
Widera-2021 of the European Union under the Grant agreement no. 101079338.

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# Looking for lepton flavour violation in supersymmetry at the LHC

Monoranjan Guchait<EMAIL_ADDRESS>Department of High Energy Physics,
Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400
005, India  Abhishek M. Iyer<EMAIL_ADDRESS>Department of
Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Colaba, Mumbai 400 005, India  Rickmoy Samanta<EMAIL_ADDRESS>Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Colaba, Mumbai 400 005, India

###### Abstract

We consider models of supersymmetry which can incorporate sizeable mixing
between different generations of sfermions. While the mixing is constrained by
the non-observation of various flavour changing neutral current (FCNC)
processes, there exist regions of SUSY parameter space where the effects of
such mixing can be probed at colliders. In this work, we explore this
possibility by focussing on the slepton sector. The sleptons are produced
through cascade decays in direct neutralino-chargino
($\chi_{2}^{0}\chi_{1}^{\pm}$) pair production at the LHC. The final state is
characterized by 3 leptons and missing energy. We probe the lepton flavour
violating (LFV) vertex originating from $\chi_{2}^{0}$ decay and identify a
distinct and unambiguous combination of the tri-lepton final state which
include a lepton pair with same flavour and same sign (SFSS) in addition to a
pair with opposite flavour and opposite sign (OFOS). This combination of tri-
lepton final state containing both OFOS and SFSS pairs, can not only suppress
the SM background but can also differentiate the flavour violating decays of
$\chi_{2}^{0}$ from its corresponding flavour conserving decays. We present
results for various signal benchmark points taking into account background
contributions assuming two luminosity options (100 fb-1 and 1000 fb-1) for LHC
Run 2 experiment.

###### pacs:

73.21.Hb, 73.21.La, 73.50.Bk

††preprint: TIFR/TH/15-12

##  I Introduction

The experiments dedicated towards the investigation of flavour physics are
considered to be one of best indirect ways to establish the existence of new
physics (NP). They play an important role in constraining the viability of
various new physics scenarios, thereby complementing the direct collider
searches. The effects which give rise to large flavour changing neutral
currents (FCNC), can also be potentially probed at the colliders. For
instance, the possibility of observing a flavour violating Higgs decay at the
Large Hadron Collider (LHC) was discussed in DiazCruz:1999xe ;
Blankenburg:2012ex ; Harnik:2012pb . Further, an observation of a
$2.5~{}\sigma$ excess in the $H\rightarrow\tau\mu$ channel by CMS CMS:2014hha
in the LHC experiment has generated a lot of interest in this sector and has
led to a plethora of analysis Pilaftsis:1992st ; Brignole:2003iv ;
Arganda:2004bz ; Azatov:2009na ; Arhrib:2012mg ; Falkowski:2013jya ; Arana-
Catania:2013xma ; Arroyo:2013kaa ; Arganda:2014dta ; Sierra:2014nqa ;
Heeck:2014qea ; Lee:2014rba ; Crivellin:2015mga ; deLima:2015pqa . The
leptonic sector in the Standard Model (SM) is also interesting owing to the
absence of FCNC. This can be attributed to the massless nature of neutrinos in
the SM. The observation of neutrino oscillations, which consequently led to a
confirmation of the massive nature of left handed neutrinos, resulting in a
non-zero decay rate for rare processes like $\mu\rightarrow e\gamma$. The
predicted branching ratio (BR) in the SM, however is negligibly small ($\sim
10^{-40}$) due to the tiny neutrino mass and is beyond the sensitivity of the
current flavour experiments. There exist several extensions of the SM which
contribute to rare processes such as $\mu\rightarrow e\gamma$ via loops,
enhancing the BR substantially to $\sim 10^{-13}-10^{-15}$ and expected to be
within the reach of the indirect flavour probes. Needless to say, an
observation of such processes is a definitive signal of the presence of
physics beyond the SM. Therefore, looking for a signal of lepton flavour
violation (LFV) directly or indirectly is a challenging avenue to find NP.
Following this argument, we explore the possibility of observing lepton
flavour violation at the LHC.

There are several models in literature which discuss the possibility of
flavour violation in the leptonic sector. In the current analysis we focus on
the supersymmetric extensions of the SM which can possess soft masses having
significant flavour mixing in the mass basis of fermions. This can lead to new
contributions to the BR of rare processes. For instance, soft masses with
flavour mixing can arise in see-saw extensions of SUSY Borzumati:1986qx ;
Hall:1985dx ; Masiero:2004js  and also inspired by SUSY GUT Masiero:2002jn ;
Masiero:2005ua ; Calibbi:2006nq ; Hirsch:2012ti ; Calibbi:2012gr .
Alternatively, introduction of flavour symmetries Feng:2007ke ; Feng:2009bd ,
models with messenger matter mixing in gauge mediated supersymmetry breaking
(GMSB)Fuks:2008ab ; Shadmi:2011hs ; Abdullah:2012tq ; Calibbi:2013mka ;
Calibbi:2014yha , models with R-symmetric supersymmetry Kribs:2007ac ;
Kribs:2009zy , supersymmetric theories in the presence of extra-spatial
dimensions Nomura:2007ap ; Nomura:2008gg ; Nomura:2008pt  etc. also lead to
flavourful soft masses. Scenarios in which mass splitting lead to flavour
violation have been considered in Galon:2013jba ; Abdullah:2012tq ;
Calibbi:2014pza . Such extensions in general lead to flavoured soft masses and
depending on the parameters can lead to observable rates for the flavour
violating decays in the squark and leptonic sector.

Flavour mixing in the sfermion mass matrices can be probed at the collider by
the flavour violating decay of a sparticle of flavour (say $i$) into a fermion
of flavour $j$ where $j\neq i$. Flavour violating decays of sleptons were
studied in the context of $e^{+}e^{-}$ linear collider Krasnikov:1994hr ;
Krasnikov:1995qq ; ArkaniHamed:1996au ; Hisano:1998wn ; Guchait:2001us ;
Carquin:2011rg ; Abada:2012re . In Ref. ArkaniHamed:1997km  the authors
studied the possibility of observing CP violation from slepton oscillations at
the LHC and NLC. At the LHC, the sleptons can be produced either through
Drell-Yan (DY) process or by cascade decays from heavier sparticles.
Subsequent flavour violating decays of sleptons produced by DY were studied in
Bityukov:1997ck ; Krasnikov:1996np  while those produced by cascade decays
were studied in Agashe:1999bm ; Hinchliffe:2000np ; Hisano:2002iy ;
Kitano:2008en ; Kaneko:2008re ; Andreev:2006sd ; Deppisch:2007rm . Probing LFV
through the measurement of splitting in the mass eigenstates of sleptons was
considered in Allanach:2008ib ; Buras:2009sg ; Galon:2011wh . In this paper we
report on our study of flavour violation in the leptonic sector by producing
sleptons in cascade decays through pair production of neutralino-chargino at
the future LHC experiments.

Starting with MSSM, we write the most general structure for the slepton mass
matrix. The constraints on the model from the non-observation of flavour
violating processes can be expressed by working in the mass-insertion
approximation (MIA) Hall:1985dx ; Gabbiani:1988rb  in terms of bounds on the
flavour violating parameter $\delta_{ij,}~{}i\neq j$ as defined in Eq.2
Gabbiani:1996hi . A non-zero $\delta_{ij}$ also opens up the possibility of
flavour violating decay as far as collider implications of flavoured slepton
masses are concerned.

Our goal is to probe the flavour violating decay in the case of first two
generations in the slepton sector in SUSY. In this context strong bounds exist
on the flavour violating parameter, coming primarily from the non-observation
of $\mu\rightarrow e\gamma$ Adam:2013mnn . There exist regions of parameter
space where these bounds can be relaxed owing to cancellations between
different diagrams contributing to this process, thereby giving access to
probe LFV at the colliders.

In this letter we explore this possibility to look for LFV decays considering
neutralino-chargino pair production in proton-proton collisions, which
eventually leads to three lepton and missing energy final state. The tri-
lepton final state is characterized by the presence of two leptons with
opposite flavour and opposite sign combination (OFOS). The presence of LFV in
the tri-lepton final state is ensured by demanding a combination of same
flavour same sign (SFSS) lepton pair along with the OFOS combination. While an
imposition of this SFSS criteria along with OFOS has a tendency to decrease
the signal, it aids in suppressing the backgrounds due to SM and SUSY
significantly.

The paper is organized as follows: In Section II we discuss the model set-up
introducing the various parameters relevant for the analysis in the framework
of a simplified model. Relevant regions of parameter space consistent with the
flavour constraints and conducive to be probed at the colliders are identified
in this section. In Section III we explain our choice of OFOS and SFSS
combination to extract the signal with a detailed description of the
simulation. The results of the simulation for the background and the
representative points for the signal events are presented. In Section IV we
show regions of the parameter space which can be probed at the LHC Run 2
experiment in the near future. We conclude in Section V.

##  II  Model Parametrization

In this section we introduce the basic model set-up and related parameters
necessary to describe LFV. In order to reduce the dependence on many
parameters, we consider a simplified SUSY model (SMS) approach with only left
handed sleptons, wino and a bino while decoupling the rest of the spectrum.
The $\mu$ term is assumed to be $\sim 1$ TeV to make the neutralino/chargino
dominantly composed of gauginos with a very small higgsino component. In this
case, the mass of $\chi_{2}^{0}$, the second lightest neutralino and
$\chi_{1}^{\pm}$, the lightest chargino, are roughly the same as $\sim M_{2}$,
the mass of the $SU(2)$ gauginos. The lightest neutralino $\chi_{1}^{0}$,
which is assumed to be the lightest supersymmetric particle (LSP) has mass
$\sim M_{1}$, same as the mass of the $U(1)$ gaugino.

For the slepton sector we focus on the flavour violation in the left handed
sector making the right handed sleptons very heavy and set the left-right
chiral mixing in the slepton mass matrix to be negligible. For simplicity, we
assume only two generations. With these assumptions, the left handed slepton
mass matrix in the basis $l_{F}\equiv(\tilde{e}_{F},\tilde{\mu}_{F})$ is given
as,

$\tilde{m}^{2}=\begin{bmatrix}m^{2}_{L_{11}}&m^{2}_{L_{12}}\\\
m^{2}_{L_{12}}&m^{2}_{L_{22}}\end{bmatrix},$ (1)

where $F$ denotes the flavour basis (SUPER CKM) for the sleptons. In this
basis the flavour violating parameter $\delta_{12}$ is parametrised as
Hall:1985dx ; Gabbiani:1988rb ,

$\delta_{12}=\frac{m^{2}_{L_{12}}}{\sqrt{m^{2}_{L_{11}}m^{2}_{L_{22}}}}.$ (2)

Naturally, this flavour violating parameter $\delta_{12}$ is coupled to the
rates corresponding to flavour violating rare decays in the first and second
generation lepton sector. Hence an upper bound on this parameter exists due to
non-observations of these rare decays like $\mu\rightarrow e\gamma$
Adam:2013mnn , $\mu-e$ conversion Wintz:1996va  and $\mu\rightarrow eee$
DeGerone:2011fg .

In order to obtain the mass eigenvalues of the sleptons, the matrix in Eq.1
can be rotated into a diagonal form by an angle $\theta$ given by,

$\sin 2\theta=\frac{2m^{2}_{L_{12}}}{m^{2}_{L_{2}}-m^{2}_{L_{1}}},$ (3)

where $m^{2}_{L_{i}}$ are the eigenvalues. It can be related to the flavour
violating parameter $\delta_{12}$ as,

$\delta_{12}=\frac{\sin 2\theta(m^{2}_{L_{2}}-m^{2}_{L_{1}})}{2m_{L}^{2}}$ (4)

where $m_{L}=\frac{m_{L_{1}}+m_{L_{2}}}{2}$. The structure of the mass matrix,
Eq.1 allows for the possibility of flavour oscillations similar to neutrino
flavour oscillations. The probability $P(\tilde{e}_{F}\rightarrow\mu)$ of a
flavour eigenstate $\tilde{e}_{F}$ decaying into a muon is given by
ArkaniHamed:1996au ,

$\displaystyle P(\tilde{e}_{F}\rightarrow\mu)$ $\displaystyle=$
$\displaystyle\sin^{2}2\theta\frac{(\Delta
m^{2})^{2}}{4\bar{m}^{2}\Gamma^{2}+(\Delta
m^{2})^{2}}BR(\tilde{\mu}\rightarrow\mu),$ (5) $\displaystyle\sim$
$\displaystyle\sin^{2}2\theta~{}BR(\tilde{\mu}\rightarrow\mu)~{}\text{for}~{}\Gamma\ll\Delta
m^{2},$

with $\Delta m^{2}=m_{L_{2}}^{2}-m_{L_{1}}^{2}$. The above expression can be
re-expressed in terms of the parameter $\delta_{12}$ from Eq.4. Thus the
branching ratio for the flavour violating decay, $\chi^{0}_{2}\rightarrow
e~{}\tilde{e}\rightarrow e~{}\mu~{}\chi^{0}_{1}$ can be computed as,

$BR(\chi^{0}_{2}\rightarrow
e~{}\mu~{}\chi^{0}_{1})=\mathcal{B}_{LFV}~{}BR(\chi_{2}^{0}\rightarrow\tilde{e}~{}e)~{}BR(\tilde{e}\rightarrow
e\chi_{1}^{0})+e\leftrightarrow\mu$ (6)

Here the suppression factor due to flavour violation is given by,

$\mathcal{B}_{LFV}=\sin^{2}2\theta=\left(\frac{m_{L}~{}\delta_{12}}{\Delta
m_{12}}\right)^{2},$ (7)

where $\Delta m_{12}=m_{L_{2}}-m_{L_{1}}$.

As mentioned before, bounds on $\delta_{12}$ and hence $\mathcal{B}_{LFV}$ can
be obtained by taking into account the experimental upper limit on the
$BR(\mu\rightarrow e\gamma)<5.7\times 10^{-13}$ Adam:2013mnn . The higher
dimensional operator contributing to this process is parametrized as
Calibbi:2015kja ,

$\mathcal{L}_{FV}=e\frac{m_{l}}{2}~{}\bar{e}~{}\sigma_{\alpha\beta}\left(A_{L}P_{L}+A_{R}P_{R}\right)~{}\mu~{}F^{\alpha\beta},$
(8)

where the model dependence is captured by the Wilson coefficients $A_{L,R}$.
The branching ratio for this process is then given by Calibbi:2015kja ,

$\displaystyle BR(\mu\rightarrow
e\gamma)=\frac{48\pi^{3}}{G_{F}^{2}}\left(|A_{L}|^{2}+|A_{R}|^{2}\right).$ (9)

In our considered model, $A_{R}\equiv 0$, as the right handed sleptons are
assumed to be very heavy. $A_{L}$ on the other hand receives three
contributions due to chargino, neutralino and bino mediated diagrams and is
given as Calibbi:2015kja ,

$\displaystyle
A_{L}=\frac{\delta_{12}}{m_{L}^{2}}\left(\frac{\alpha_{Y}}{4\pi}f_{n}\left(\frac{M_{1}^{2}}{m_{L}^{2}}\right)+\frac{\alpha_{Y}}{4\pi}f_{n}\left(\frac{M_{1}^{2}}{m_{L}^{2}}\right)+\frac{\alpha_{2}}{4\pi}f_{c}\left(\frac{M_{2}^{2}}{m_{L}^{2}}\right)\right)$
(10)

where $f_{n,c}$ are loop factors defined in Calibbi:2015kja  with a non-
trivial mass dependence of related sparticles and $\alpha_{Y},\alpha_{2}$ are
the $U(1)_{Y}$ and $SU(2)$ gauge couplings.

The analysis can be simplified again by choosing the following parametrization
for the mass $M_{1}$ of the (LSP) $\chi_{1}^{0}$,

$M_{1}=\frac{M_{2}}{2},$ (11)

which is the relation at the electroweak scale due to unification of gaugino
masses at the GUT scale. For the sleptons we choose,

$M_{2}>m_{L}>M_{1}.$ (12)

This relation assumes that the intermediate sleptons in $\chi_{2}^{0}$ decay
are produced on-shell by requiring that they are lighter than the mass of
$\chi^{0}_{2}$ $\simeq M_{2}$. Under these assumptions, we try to find the
available range of parameters allowed by existing $\mu\rightarrow e\gamma$
constraints as will be discussed later. Fig.1 shows the region in the
$M_{2}-m_{L}$ plane for which the conditions in Eq.11 and 12 are satisfied
(green region). It depicts the region of parameter space which is of interest
as far as collider implications are concerned as discussed in this paper.

The blue region shows the parameter space for which $BR(\mu\rightarrow
e\gamma)<5.7\times 10^{-13}$ is satisfied for $\delta_{12}=0.01$ in the left
plot and for $\delta_{12}=0.02$ in the right plot. As expected, due to the
smaller value of $\delta_{12}$, the blue region in the left plot has a larger
overlap with the green region as compared to the plot in the right, thereby
admitting smaller slepton masses. The orange region in both the plots shows
the parameter space for which $BR(\mu\rightarrow e\gamma)<5.7\times 10^{-13}$
is satisfied for $\delta_{12}=0.1$. We find that there is virtually no overlap
with the region which is of interest to us from the view of collider searches.

|
---|---
Figure 1:  Region satisfying Eq.11 and 12 (green), while the orange regions
satisfy the $\mu\rightarrow e\gamma$ constraint for $\delta_{12}=0.1$. The
blue regions are allowed by the upper bound on BR($\mu\rightarrow e\gamma$)
for $\delta_{12}=0.01$ (left) and $\delta_{12}=0.02$ (right). Units of mass
are in GeV.

It would be interesting to estimate the suppression factor $\mathcal{B}_{LFV}$
corresponding to the allowed region in the $M_{2}-m_{L}$ plane for the values
of $\delta_{12}$ in Fig.1. As seen in Eq.7, the parameter $\mathcal{B}_{LFV}$,
which determines the rate for LFV, is sensitive to the mass-splitting $\Delta
m=m_{L_{2}}-m_{L_{1}}$ and $m_{L}$. $\mathcal{B}_{LFV}$ increases with
$\delta_{12}$ which can only be accommodated with a larger $m_{L}$. Thus
smaller values of $\delta_{12}$ are not conducive to generate a large
$\mathcal{B}_{LFV}$. $\mathcal{B}_{LFV}$ is also inversely proportional to the
mass splitting $\Delta m$. However, it cannot increase indefinitely as
$\mathcal{B}_{LFV}\leq 1$, leading to a lower bound on $\Delta m$. Fig.2
demonstrates the contours of constant $\mathcal{B}_{LFV}$ in the $\Delta
m-m_{L}$ plane. We find that for $\delta_{12}=0.02$, slepton in excess of 250
GeV are required to get $\mathcal{B}_{LFV}\geq 0.1$, while being consistent
with the flavour constraints (overlap of blue and green region) in Fig. 1.

|
---|---
Figure 2: Contours of $\mathcal{B}_{LFV}$ for $\delta_{12}$=0.01 (left) and
$\delta_{12}=0.02$ (right). The horizontal blue line is excluded by
$BR(\mu\rightarrow e\gamma)$ for $\delta_{12}=0.01$ (left) and
$\delta_{12}=0.02$ (right). The units of mass are in GeV.

##  III Signal and Background simulations

As mentioned in the introduction, we probe the signal of LFV in slepton decay
producing it via cascade decays of sparticles which are produced in proton-
proton collisions at the LHC. Here we focus on $\chi_{1}^{\pm}\chi_{2}^{0}$
production which eventually leads to a tri-lepton final state as,

$\displaystyle pp\rightarrow\begin{cases}\chi_{2}^{0}\rightarrow
l_{i}^{\pm}\tilde{l}_{i}^{\mp}\rightarrow
l_{i}^{\pm}l_{j}^{\mp}\chi_{1}^{0},\;\;\;i\neq j,\\\ \chi_{1}^{\pm}\rightarrow
l_{i}^{\pm}\nu\chi_{1}^{0},\end{cases}$ (13)

where $i,j$ denote flavour indices ($e,\mu$). The flavour violating vertex
causes the decay of a slepton ($\tilde{l}_{i}$), coming from $\chi_{2}^{0}$
decay, in Eq.13, into a lepton of flavour $l_{j}$ with $i\neq j$. It is clear
from the above process that the signature of LFV is the presence of 3 leptons
of which 2 leptons are with opposite flavour and opposite sign (OFOS) in
addition to missing energy ($\not{E}$) due to the presence of two LSP and
neutrino. The leptons with OFOS originate from $\chi^{0}_{2}$ decay while the
third lepton comes from the $\chi^{\pm}_{1}$ decay. Thus, following this decay
scenario, it is possible to have 8 combinations of tri-leptons, each having at
least one OFOS lepton pair as,

$\displaystyle
e^{+}e^{+}\mu^{-}~{};e^{-}e^{-}\mu^{+}~{};\mu^{-}e^{+}\mu^{-}~{};\mu^{+}e^{-}\mu^{+}$
$\displaystyle
e^{+}e^{-}\mu^{+}~{};e^{-}e^{+}\mu^{-}~{};\mu^{+}e^{+}\mu^{-}~{};\mu^{-}e^{-}\mu^{+}.$
(14)

On the other hand, the pair production of $\chi^{\pm}_{1}\chi_{2}^{0}$ will
also give rise to tri-lepton final state with a flavour conserving decay of
$\chi_{2}^{0}$ i.e. $\chi_{2}^{0}\rightarrow l^{+}l^{-}\chi_{1}^{0}$. Note
that this flavour conserving decay scenario also results in 8 combinations of
tri-lepton final state given as

$\displaystyle
e^{+}\mu^{+}\mu^{-}~{};e^{-}\mu^{+}\mu^{-}~{};\mu^{-}e^{+}e^{-}~{};\mu^{+}e^{+}e^{-}$
$\displaystyle\mu^{+}\mu^{+}\mu^{-}~{};e^{-}e^{+}e^{-}~{};e^{+}e^{+}e^{-}~{};\mu^{-}\mu^{-}\mu^{+}$
(15)

out of which four combinations of OFOS exist as seen in the first line of
Eq.15. It is clearly a potential background corresponding to the signal
channel in Eq.14 and expected to have the same rate as signal. However, a
closer look at these two final states in Eq.14 and 15 reveals a characteristic
feature. For example, in the case of signal, out of the 8 combinations of tri-
leptons with OFOS combinations, notice that four combinations shown in the
first line in Eq.14, also possess a pair of leptons with same flavour and same
sign (SFSS) which are absent in the background final states, shown in Eq.15.
The rest of the states with OFOS combination in Eq.14 are identical to the
final states given in Eq.15. We exploit this characteristic feature to extract
the LFV signal events out of all three lepton events including all
backgrounds. Thus our signal is composed of three leptons having combinations
of both OFOS and SFSS together, which is an unambiguous and robust signature
of LFV in SUSY. Note that while choosing a clean signature of LFV decay in
SUSY, we pay a price by a factor of half as is clear from Eq.14. However, this
specific choice of combinations in tri-leptons is very powerful in eliminating
much of the dominant SM backgrounds arising from $WZ$ and $t\bar{t}$ following
leptonic decays of $W/Z$ and top quarks.

We now discuss our simulation strategy to estimate the signal rates while
suppressing the SM and SUSY backgrounds. We performed simulations for both
signal and background using  PYTHIA8 Sjostrand:2007gs  at 14 TeV centre of
mass energy and applying the following selections:
$\bullet$ Jet selection: The jets are reconstructed using FastJet
Cacciari:2011ma  and based on anti $k_{T}$ algorithm Cacciari:2008gp  setting
the jet size parameter $R=0.5$. The jets passing the cuts on transverse
momentum $p_{T}^{j}\geq 30{\rm~{}GeV}$, pseudo-rapidity $|\eta^{j}|\leq 3.0,$
are accepted.
$\bullet$ Lepton selection: Our signal event is composed of three leptons and
are selected according to the following requirement on their transverse
momenta and the pseudo-rapidity: $p_{T}^{\ell_{1,2,3}}\geq
20,20,10{\rm~{}GeV};~{}|\eta^{\ell_{1,2,3}}|\leq 2.5,$ where the leptons are
$p_{T}$ ordered with $p_{T}^{\ell_{1}}$ being the hardest one. In addition,
the leptons are also required to be isolated i.e. free from nearby hadronic
activities. It is ensured by requiring the total accompanying transverse
energy, which is the scalar sum of transverse momenta of jets within a cone of
size $\Delta R(l,j)\leq 0.3$ around the lepton, is less than 10$\%$ of the
transverse momentum of the corresponding lepton.
$\bullet$ Missing transverse momentum: We compute the missing transverse
momentum by carrying out a vector sum over the momenta of all visible
particles and then reverse its sign. Since $\not{p_{T}}$ is hard in signal
events, so we apply a cut $\not{p_{T}}\geq 100$ GeV.
$\bullet$ Z mass veto: We require that in three lepton events, the invariant
mass of two leptons with opposite sign and same flavour should not lie in the
mass window $m_{ll}=M_{Z}\pm 20$ GeV. It helps to get rid of significant
amount of $WZ$ background.
$\bullet$ $b$ like jet selection: The $b$ jets are identified through jet-
quark matching i.e. those jets which lie with in $\Delta R(b,j)<0.3$ are
assumed to be $b$ like jets.
$\bullet$ OFOS: Our signal event is characterised by the requirement that it
has at least one lepton pair with opposite flavour and opposite sign.
$\bullet$ SFSS: We require the presence of SFSS combination along with OFOS
combination in three lepton final state, which is the characteristic of our
signal. As stated before, this criteria is very effective in isolating the
background due to the same SUSY process but for their subsequent flavour
conserving decays, in particular for $\chi_{2}^{0}$ decay.

We perform our analysis by choosing various representative points in the SUSY
parameter space. The spectrum is generated using  SUSPECT Djouadi:2002ze  and
the decays of the sparticles are computed using SUSYHIT Djouadi:2006bz . Table
1 presents the six representative points (A-F) for which we discuss the
details of our simulation. From A to F, the spectrum is characterized by
increasing masses of gauginos, with the slepton mass $m_{L}$ lying midway
between the two, $m_{L}=\left(M_{1}+M_{2}\right)/2$.

Spectrum Characteristics | A | B | C | D | E | F
---|---|---|---|---|---|---
$\chi_{2}^{0}/\chi^{\pm}_{1}$ | 210 | 314 | 417 | 518 | 619 | 718
$\chi_{1}^{0}$ | 95.8 | 144 | 193 | 241 | 290 | 339
$m_{L}$ | 156 | 229 | 303 | 377 | 452 | 526
$BR(\chi^{0}_{2}\rightarrow\tilde{e}_{L}e)$ | 0.13 | 0.15 | 0.16 | 0.16 | 0.16 | 0.16
$BR(\chi^{0}_{2}\rightarrow\tilde{\mu}_{L}\mu)$ | 0.13 | 0.15 | 0.16 | 0.16 | 0.16 | 0.16
Table 1: Representative choices of SUSY parameter space. All masses are in
GeV.

In Table 2 and 3 we present the effects of selection of cuts in simulation for
both the signal and backgrounds respectively. In addition to the SM
backgrounds which are mainly due to $t\bar{t}$ and $WZ$, we also simulate the
background taking into account the contributions due to flavour conserving
decay of $\chi_{2}^{0}$ for each of the representative points in Table 1.
There are other sub-dominant backgrounds like $tbW$, $ZZ$ if one lepton is
missed or $WW$, if jets fake as leptons. However these backgrounds are
expected to be very small and not considered here. We present results for
signals corresponding to those representative parameter space as shown in
Table 2 . In this table, the first column shows the sequence of cuts applied
in the simulation, while the second column onwards event yields for the signal
are shown. Table 3 presents the same for the backgrounds due to SUSY in the
second column and the SM in the third column. Notice that lepton isolation
requirement and a cut on $\not{p_{T}}$ has considerable impact in reducing
$t\bar{t}$ and $WZ$ background. As noted earlier, we find the SFSS criteria to
be very effective in isolating the SUSY background due to flavour conserving
decay of $\chi_{2}^{0}$ for all the representative points in Table 1 .
Finally, it is possible to have large number of tri-lepton events in
background processes, but imposition of specific choices like OFOS and SFSS
along with large missing energy cut help in isolating it to a great extent as
shown in Table 3 . In spite of this suppression of background events, the
signal yields are far below than the total background contribution owing to
it’s huge production cross sections as shown in Table 3. Therefore, in order
to improve signal sensitivity further, we impose additional requirements by
looking into the other characteristics of signal events. For example, signal
events are free from any kind of hadronic activities at the parton level i.e.
no hard jets are expected in the signal final state, whereas in background
process, in particular events from $t\bar{t}$ are accompanied with large
number of jets. We exploit this fact to increase signal sensitivity by adding
following criteria.

| Signal($\chi_{2}^{0}\chi_{1}^{\pm}$)
---|---
$M_{2}\Longrightarrow$ | 200 | 300 | 400 | 500 | 600 | 700
No. of events generated | 10000 | 10000 | 10000 | 10000 | 10000 | 10000
$p^{\ell_{1,2}}_{T}>20,p^{\ell_{3}}_{T}>10,|\eta|<2.5$ | 1371 | 1752 | 2014 | 2218 | 2225 | 2342
Lepton isolation cut | 1330 | 1669 | 1883 | 2055 | 2036 | 2112
$\not{p_{T}}>100$ | 474 | 959 | 1326 | 1600 | 1683 | 1860
OFOS | 470 | 952 | 1319 | 1581 | 1659 | 1828
Z mass veto | 423 | 849 | 1218 | 1485 | 1574 | 1752
SFSS | 223 | 462 | 640 | 783 | 804 | 892
Case a: jet veto | 91 | 205 | 288 | 337 | 346 | 380
Case b: $b$-like jet veto | 221 | 458 | 635 | 777 | 798 | 884
Case c: $n_{j}\leq 1$ and $b$-like veto | 161 | 375 | 479 | 604 | 617 | 687
Table 2: Event summary for signal after all selections. All energy units are in GeV.  | SUSY($\chi_{2}^{0}\chi_{1}^{\pm}$) | SM
---|---|---
| A | B | C | D | E | F | $t\bar{t}$ | WZ
$M_{2}\Longrightarrow$ | 200 | 300 | 400 | 500 | 600 | 700 | - | -
Cross section ($fb$) at 14 TeV | $1.65\times 10^{3}$ | 370.5 | 118.8 | 45.6 | 20.5 | 9.57 | $9.3\times 10^{5}$ | $4.47\times 10^{4}$
No. of events generated | 10000 | 10000 | 10000 | 10000 | 10000 | 10000 | $10^{7}$ | $3\times 10^{6}$
$p^{\ell_{1,2}}_{T}>20,p^{\ell_{3}}_{T}>10,|\eta|<2.5$ | 1299 | 1779 | 2015 | 2195 | 2245 | 2361 | 164895 | 23960
Lepton isolation cut | 1251 | 1672 | 1874 | 2044 | 2051 | 2131 | 70233 | 22366
$\not{p_{T}}>100$ | 454 | 967 | 1311 | 1624 | 1722 | 1872 | 19241 | 1669
OFOS | 209 | 482 | 656 | 820 | 855 | 918 | 14012 | 858
Z mass veto | 126 | 346 | 547 | 728 | 768 | 853 | 12395 | 122
SFSS | 4 | 6 | 11 | 14 | 15 | 25 | 4598 | 22
Case a: jet veto | $\leq 1$ | 1 | 1 | 5 | 4 | 4 | 29 |  $\leq$ 1
Case b: $b$-like jet veto | 4 | 5 | 10 | 14 | 13 | 23 | 131 | 13
Case c: $n_{j}\leq 1$ and $b$-like veto | 1 | 3 | 7 | 9 | 9 | 19 | 48 | 5
Table 3: Event summary for SUSY and SM background. All energy units are in
GeV.

Case a: Jet veto
In this case we reject events if it contain any hard jets. In Table 3 we see
that while the jet veto criteria reduces the $t\bar{t}$ and WZ background
significantly, but it also substantially damage the signal by a factor of 2 or
3 as shown in Table 2 . In signal process, jets arise mainly from the hadronic
radiation in initial and final states and it is true for all the
representative signal points. The reason can be attributed to enhancement of
hadronic activities at higher energies. Nevertheless the jet veto seems to be
useful to improve signal to background ratio. However we consider two more
alternatives with a goal to increase signal sensitivity further:

Case b: $b$-like jet veto
Here we eliminate events if there be at least one $b$ like jet. As can be seen
Table 3, $b$ jet veto is more efficient than the jet veto condition, as the
$t\bar{t}$ background is suppressed by a few orders of magnitude without
costing the signal too much.

Case c: Apply $b$-like jet veto and number of jets $n_{j}\leq 1$
Here we apply the $b$-like jet veto condition along with the presence of
maximum one jet. As seen in Table 3 , it is very helpful in reducing the
$t\bar{t}$ background significantly but it does not affect the signal as much
as the simple jet veto condition ($n_{j}=0$) does alone.

Note that we have identified $b$-like jets by a naive jet-quark matching which
is an overestimation from the realistic b-jet taggingChatrchyan:2012jua  which
is out of scope of the present analysis. However, for the sake of
illustration, we present these results with b-jet veto, (case (b) and (c)), to
demonstrate that this criteria might be very useful in suppressing
backgrounds, which requires more detector based simulation. In view of this,
we focus only on the results obtained by using jet veto, case(a) for further
discussion.

We also present the dilepton $(e\mu)$ invariant mass distributions for the
spectrum A (left) and F (right) in Fig.3 normalizing it to unity. It is
subject to all primary selection cuts on leptons and jets, including the OFOS
and SFSS combination. The $m_{e\mu}$ distribution is expected to have a sharp
edge on higher side, which can be derived analytically from kinematical
consideration. The position of this edge of $m_{e\mu}$ is given as
Hisano:2002iy ; Gjelsten:2004ki ,

$\displaystyle(m^{max}_{e\mu})^{2}=m_{\chi_{2}^{0}}^{2}\left(1-\frac{m_{L}^{2}}{m_{\chi_{2}^{0}}^{2}}\right)\left(1-\frac{m_{\chi_{1}^{0}}^{2}}{m_{L}^{2}}\right).$
(16)

The appearance of an edge in the $m_{e\mu}$ distribution is a clear indication
of LFV vertex in the $\chi^{0}_{2}$ decay. However, this $m_{e\mu}$
distribution is affected by a combinatorial problem. For each tri-lepton
event, two OFOS pairs can be constructed: a) both leptons coming from
$\chi^{0}_{2}$ decay and b) “imposter” pair with one lepton from
$\chi^{0}_{2}$ and the other from $\chi^{\pm}_{1}$. In Fig.3 the red (dotted)
curve represents the dilepton invariant mass distribution of the leptons
tracked to the $\chi^{0}_{2}$ vertex while blue (solid) curve corresponds to
dilepton without any prior information about their origin. It (red dotted
line) exhibits a very distinct edge as the identity of the lepton pair
originating for $\chi^{0}_{2}$ is known a-priori. The (solid) blue line is
more realistic as it includes both the correct OFOS and SFSS pair as well as
the contamination due to the “impostor” pair which is responsible for a tail
beyond the edge. As a result it exhibits a more diffused behaviour near the
position of the edge. However, we can roughly estimate the position of the
edge using the blue (solid) line as $\sim 120$ GeV for the left panel and
$\sim 375$ GeV for the right panel. We find that these values are in fairly
good agreement with the corresponding numbers used in our simulation. It may
be noted here that such distributions with a sharp edge are the characteristic
feature of these type of decays which can also be exploited to suppress
backgrounds Hisano:2002iy  in order to increase signal to background ratio.

|
---|---
Figure 3: OFOS dilepton invariant mass distribution for spectrum A (left) and spectrum F (right). The events are selected at the SFSS level. | Signal (S) | Background (B)
---|---|---
Properties | A | B | C | D | E | F | $t\bar{t}$ | WZ
Cross section ($fb$) at 14 TeV | $1.65\times 10^{3}$ | 370.5 | 118.8 | 45.6 | 20.5 | 9.57 | $9.3\times 10^{5}$ | $4.47\times 10^{4}$
Normalized cross sections
Case a: jet veto | 15.01 | 7.59 | 3.41 | 1.51 | 0.67 | 0.37 | 2.69 | $\leq 1$
Case b: $b$-like veto | 36.4 | 16.9 | 7.54 | 3.54 | 1.63 | 0.85 | 12.1 | 0.19
Case c: $n_{j}\leq$ 1 and $b$-like veto | 26.5 | 13.9 | 5.7 | 2.75 | 1.26 | 0.66 | 4.4 | 0.07
$\frac{S}{\sqrt{B}}(@100)$ fb-1
Case a: jet veto | 91.43 | 45.93 | 20.78 | 9.32 | 4.31 | 2.24 | - | -
Case b: $b$-like veto | 100.99 | 47.87 | 21.34 | 10.04 | 4.64 | 2.43 | - | -
Case c: $n_{j}\leq$ 1 and $b$-like veto | 122.4 | 64.4 | 26.4 | 12.8 | 5.92 | 3.12 | - | -
Table 4: Normalized cross-section ($fb$) and $S/\sqrt{B}$ for signal and
background subject to three selection conditions

##  IV Results and Discussions

Table 4 gives the normalized signal and background cross-sections due to all
selection cuts. These are obtained by multiplying the production cross section
given in the first row by acceptance efficiencies. The production cross
section are estimated by multiplying the leading order (LO) cross section
obtained from PYTHIA8 with the corresponding $k$ factors 111The appropriate
$k$ factors for $t\bar{t}$ and WZ processes is 1.6 Kidonakis:2011tg  and 1.7
Campbell:2011bn  respectively while for the signal it is 1.5 Beenakker:1999xh
. . Corresponding to these signal and background cross sections, we also
present the signal significance by computing $S/\sqrt{\rm B}$ for integrated
luminosity 100 fb-1 as shown in the bottom of Table 4. Although case(b)
corresponding to $b$-like jet veto results in the largest cross section for
all signal parameter space, signal significance does not improve due to
comparatively less suppression of SM backgrounds. With the increase of gaugino
masses acceptance efficiencies goes up as final state particles become
comparatively harder, but $S/\sqrt{B}$ is depleted due to drop in
$\chi_{2}^{0}\chi_{1}^{\pm}$ pair production cross-section. While estimating
signal rates and significance, we assume a maximal flavour violation i.e.
$\mathcal{B}_{LFV}=1$. Obviously, a further suppression is expected by a
factor $\mathcal{B}_{LFV}$ which depletes the BR of $\chi_{2}^{0}$, (see
Eq.6). For a given $\delta_{12}$, $\mathcal{B}_{LFV}$ is a function of the
slepton mass as well as the mass splitting $\Delta m$ as shown in Fig. 2. For
instance $S/\sqrt{B}$ may suffer by an order of magnitude for
$\mathcal{B}_{LFV}=0.1$. While the lower end of the spectrum can lead to a
larger $S/\sqrt{B}$, the corresponding $\mathcal{B}_{LFV}$ decreases as we
move further towards the IR part of the slepton spectrum. This can be
attributed to stronger bounds on $\delta_{12}$ for lower slepton masses.
Though the lower mass is not yet ruled out, it is more economical to consider
relatively heavier slepton masses as the bounds from current and future
experiments will be relatively weaker.

In Fig.4, we illustrate this mass sensitivity by presenting $S/\sqrt{B}$
obtained using jet veto condition case (a). Notice that for a given
$\chi_{1}^{\pm}$ and $\chi_{2}^{0}$ masses, signal is not very sensitive to
slepton mass as long as it is produced on-shell from $\chi_{2}^{0}$ decay and
$M_{\chi_{2}^{0}}-m_{L}$ is sufficiently high. The regions in the
$M_{2}-m_{L}$ plane correspond to different values of $S/\sqrt{B}$ computed
for $\mathcal{L}=100$ fb-1 and by assuming $\mathcal{B}_{LFV}=1$. The sleptons
and gaugino masses follow the parametrisation in Eq.11 and 12. It is
superimposed on the region satisfying $BR(\mu\rightarrow e\gamma)<5.7\times
10^{-13}$ for $\delta_{12}=0.01$ (left) and $\delta_{12}=0.02$ (right). As
seen from Table 4 and Fig. 4, the signal significance is better for lower
masses due to larger $\chi_{2}^{0}\chi_{1}^{\pm}$ pair production cross
section. However, it suffers by smaller values of $\mathcal{B}_{LFV}$
corresponding to those slepton masses as shown in Eq.7 and Fig.2.

|
---|---
Figure 4: Variation of $S/\sqrt{B}$ (using jet veto, case (a) for different
regions with two choices of $\delta_{12}=0.01$ (left) and
$\delta_{12}=0.02$(right). The regions light blue are allowed by
BR($\mu\rightarrow e\gamma$) constraint. Here we assume $\mathcal{B}_{LFV}=1$.
Masses are in GeV.

Fig.5 shows the sensitivity reach of $\mathcal{B}_{LFV}$ in the $M_{2}-m_{L}$
plane using the parametrisation in Eq.11 and 12. The numbers in boxes for
different coloured regions give the minimum values of $\mathcal{B}_{LFV}$
which can be probed, while requiring a $5\sigma$ discovery corresponding to
those values of $M_{2}$ and $m_{L}$ and are presented for two different
options of luminosities: $\mathcal{L}=100~{}fb^{-1}$ (left) and
$\mathcal{L}=1000~{}fb^{-1}$ (right). As the constraints from indirect flavour
measurements get tighter, larger $\mathcal{B}_{LFV}$ can be attained with
heavier slepton masses, while respecting bounds from the rare decays as shown
in Fig. 1 and 2. For example, for lower masses
$\chi_{2}^{0}\sim\chi^{\pm}_{1}\sim 250$ GeV and $m_{L}\sim 200$ GeV, the LFV
parameter $\mathcal{B}_{LFV}\sim 0.05$ or more can be probed at $5\sigma$
level of signal sensitivity for $\mathcal{L}=100~{}fb^{-1}$ . As expected, the
minimum $\mathcal{B}_{LFV}$ required for a $5\sigma$ sensitivity goes up,
thereby reducing the sensitivity of $\mathcal{B}_{LFV}$ measurement with the
increase of gaugino and slepton masses and this can be attributed to the drop
in cross-sections. The left plot in Fig.5 is terminated at the point
corresponding to a requirement of $B_{LFV=1}$ for a $5\sigma$ discovery. As a
result, the representative points E and F corresponding to heavier slepton
masses are beyond the sensitivity of LHC at $\mathcal{L}=100~{}fb^{-1}$ as
they require $\mathcal{B}_{LFV}>1$ to achieve a $5\sigma$ discovery. However,
flavour violating decays with heavier slepton masses as high as 650 GeV can be
probed with an integrated luminosity of $\mathcal{L}=1000~{}fb^{-1}$ as shown
in the right plot of Fig 5.

|
---|---
Figure 5: Minimum value (in small box) of $\mathcal{B}_{LFV}$ for a
$S/\sqrt{B}=5$ discovery for $\mathcal{L}=100~{}fb^{-1}$ (left) and
$\mathcal{L}=1000~{}fb^{-1}$ (right). The $S/\sqrt{B}$ is computed using jet
veto condition. The filled triangles correspond to the representative points
A-F from left to right. The plot is truncated at the point where
$\mathcal{B}_{LFV}>1$ is required to get a $5~{}\sigma$ sensitivity of signal
for that particular luminosity. Masses are in GeV.

##  V Conclusion

The observation of flavour violating rare decays would be one of the best
indicators of the existence of physics beyond the SM. Measurements of such
decays play an important role in constraining several new physics models and
hence has received a lot of attention recently. We attempt to explore the
flavour violation in the lepton sector in the context of well motivated models
of flavourful supersymmetry. We follow an approach based on a simplified model
with only the left handed sleptons along with the neutralinos which are
gaugino dominated. We consider pair production of $\chi^{0}_{2}\chi^{\pm}_{1}$
and their subsequent leptonic decays which includes the LFV decays of
$\chi_{2}^{0}$. The final state is composed of three leptons and accompanied
by large missing energy. In addition to the presence of a lepton pair with
OFOS, we observed that certain tri-lepton combinations are also characterized
by a lepton pair with SFSS -which is a unique and robust signature of LFV in
SUSY.

The discovery potential of observing this LFV signal is primarily dependent on
the masses of sleptons and gauginos. These masses are however constrained by
non-observation of FCNC decays such as $\mu\to e\gamma$ and they get stronger
as the flavour violating parameter $\delta_{12}$ becomes larger. We have
identified the allowed range of slepton and gaugino masses relevant for our
study. In addition variation of LFV parameter ${\cal B}_{LFV}$ with masses of
slepton and mass difference between lepton mass eigenstates ($\Delta m$) are
also presented.

Estimating the various background contributions, we predict the signal
sensitivity for a few representative choices of SUSY parameters. The
combination of three leptons with OFOS and SFSS is found to be very useful to
achieve a reasonable sensitivity. It is found that for gaugino masses $\sim$
250 GeV and slepton masses $\sim$200 GeV, the LFV parameter ${\cal B}_{LFV}$
as low as 0.05 can be probed with 100 fb-1 integrated luminosity. For heavier
masses $\sim 600-700$ GeV, because of reduced $\chi_{2}^{0}\chi_{1}^{\pm}$
pair production cross section, the measurement of LFV parameter ${\cal
B}_{LFV}$ requires higher luminosity $\sim$1000 fb-1. Our study clearly
establishes the prospects of finding LFV signal in this SUSY channel at the
LHC Run 2 experiment with high luminosity options.

##  VI Acknowledgements

AI and RS would like to thank Debjyoti Bardhan and Sanmay Ganguly for
discussions.

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