# Gaussian Smoothen Semantic Features (GSSF) - Exploring the Linguistic
Aspects of Visual Captioning in Indian Languages (Bengali) Using MSCOCO
Framework

Chiranjib Sur
Computer & Information Science & Engineering Department, University of
Florida.
Email<EMAIL_ADDRESS>
###### Abstract

In this work, we have introduced Gaussian Smoothen Semantic Features (GSSF)
for Better Semantic Selection for Indian regional language-based image
captioning and introduced a procedure where we used the existing translation
and English crowd-sourced sentences for training. We have shown that this
architecture is a promising alternative source, where there is a crunch in
resources. Our main contribution of this work is the development of deep
learning architectures for the Bengali language (is the fifth widely spoken
language in the world) with a completely different grammar and language
attributes. We have shown that these are working well for complex applications
like language generation from image contexts and can diversify the
representation through introducing constraints, more extensive features, and
unique feature spaces. We also established that we could achieve absolute
precision and diversity when we use smoothened semantic tensor with the
traditional LSTM and feature decomposition networks. With better learning
architecture, we succeeded in establishing an automated algorithm and
assessment procedure that can help in the evaluation of competent applications
without the requirement for expertise and human intervention.

###### Index Terms:

image description, language translation, visual captions, semantic learning,
Bengali captions.

##  I Introduction

Image captioning [1] applications have gained massive attention from the
research community both in academia and industry due to its ability to find a
confluence of media and language and create the capability for machines to
communicate in different languages with the external world. Image captioning
has a wide range of applicability to tag enormous resources in the form of
images [48] and video, and vast amounts of new media contents are added daily,
which are un-categorized, and no one knows what its content is. This kind of
image captioning brings these media in the mainstream web system through the
use of these machines learned and automated infrastructures and applications.
While general object detection models introduced huge improvement like ResNet
([51, 70]), GoogleNet [44], Vgg [48], through the use of ImageNet [11] data
sources, there is need to establish the linkage and interaction among the
different objects [37] and establish in sentences. While a large part of the
resources is available in English languages, many regional languages are
neglected, while large investments for these languages are not viable options.
While image captioning is a complex task and gathering improvement is
difficult, mainly because machines still do not get acquainted with the
language attributes [70] and grammar [44], we experiment with the viability of
machine translation as an alternative. However, the use of machines is not to
outsource the best practices of literary works, but to convey and interpret
the messages without ambiguity. We discuss different existing image captioning
architectures and translation procedures in the subsequent sections. Previous
works in image captioning were relied on object and attribute detectors to
describe images ([48, 50]) and later mostly concentrated on attention ([46,
44, 57, 51, 35, 41, 12]) and compositional characteristics [70] and top-down
compositions [37]. At the same time, there were limited works on feature
refinement and redefining the exiting subspace with more effective and
efficient ones. [3] claimed to provide template-based sentence fill up, but no
details are provided for the criteria of how objects are selected for
templates. We argue that sentence generation can be driven only through
semantic and proper selection of the semantic layer composition and can
enhance the accurate description of the images.

In this work, we have mainly concentrated on the viability of the other
language-based image captioning systems. While English is a structural
language (syntactic and grammatical), there are unstructured languages (only
grammatically) like most of the Indian languages, which are derived from
Sanskrit like Bengali (90% Sanskrit vocabulary), Tamil (70% Sanskrit
vocabulary), Hindi (50% Sanskrit vocabulary), etc. to name a few. We will be
mainly focusing on the Bengali language as a case study and demonstrate how
they do for the image captioning counterparts. We have used the Google
Translation API (googletrans.Translator) for our work. Though the Bengali
language is unstructured, the captioning model performed well for generating
the captions. However, this is the first of its kind, never tried research-
topic. Due to the absence of any such dataset, we have used a translation
application for data generation. It is justified that such a model can be
built for a language like Bengali, as it is the fifth-largest widely spoken
language in the world and is one of the primitive languages of the world. We
have also demonstrated that if we provide a Gaussian smoothening of the
semantic layer, we can deliver better captions.

Figure 1: Overview of The Image Description Generation Methodology Using
Google Translation (Google Translation API (googletrans.Translator)). Figure
2: Example Instances of Generated Translated Data Along with English
References

###  I-A Image Caption Problem

The problem of image caption [1] is rooted in understanding the network,
relationship, interaction [74] of the different objects and descriptions.
These connections are evident to humans but it had to be taught to machines to
be able to leverage from these visual features. It is difficult to generalize
for different images if the sentences are generated from the object
combinations through the determination of the sequentiality and priority of
these objects, it is an NP-Hard problem. The models are made to determine the
region and the objects in the image heuristically and then the description. It
is said to be more than object detection and object subcategory or details
interpretation. Image captioning in regional languages require immense
exploration with the increasing number of applications expressing with images.
Also, a large part of the Indian subcontinent is still dependent on regional
language as their source of expression. These regional language-based
expressions comprise a form of art as these applications integrate well and
get constantly improved through human intervention. Image captioning in
different Indian languages has its challenges, where each of these languages
possesses a different set of language attributes, grammatical properties, and
completely different from English and other European languages. These Indian
languages are entirely different from their root language Sanskrit. The
diversification of these languages was created through different oral
transformations, which was evident in ancient India due to the absence of
texts and learning centers for everyone. Languages and customs propagated
through “Smriti” (or oral propagation) through different people, which later
transferred to other parts of the world, including Europe. This fact is
evident from the common words like “Matri” became Mother, “Pitri” became
Father, “Brahta” became Brother, “Jamity” become Geometry and even “Om” became
Amen. However, from the prospect of grammar and way of construction, Indian
languages are far different from Latin-originated languages, and in this work,
we made an effort to generate sentences in these primitive and ancient culture
languages.

The rest of the document is arranged with revisit of the existing works in
literature in Section LABEL:section:literature, the description and statistics
of the data in Section II, learning network and representation detailed
description in Section III, the intricacies of our methodology in Section IV,
experiments, results and analysis in Section V, concluding remarks with future
prospects in Section VI,

The main contribution of this work: 1) Bengali (Indian) Language Caption
Generators 2) Framework for Translator based training 3) Gaussian Smoothen
Semantic Features for Better Semantic Selection 4) Compositional-
Decompositional Network Framework for better representation for captions.

##  II Reference Exploitation & Google Translate

There are around 31 official languages, while the constitution recognizes 22
of these languages. As a result, every such language provides a different
prospect of these images, and the development of machine generation in these
languages provide better solutions to many applications, including automatic
communicator, regional aid provider, and the list is endless. In this work, we
made an effort to develop a caption generator and generated a training dataset
of captions from the existing English MSCOCO dataset through the use of Google
Translation API (googletrans.Translator). While it is true that all the
translated captions are not perfect, but it can, at least, provide a good
starting point and provide some evaluation standard.

TABLE I: Statistics of Training Data for Different Indian Languages. Statistical Estimation of Vocabulary Size and Vocabulary Appearance Provides Insights into the Capability of Networks to Learn Language | Population | Vocabulary
---|---|---
English | 8791 | 8K
Bengali | 9185 | 20K

Table I provided an estimation of the vocabulary sizes, while Figure 2
provided some instances of the translated captions in different languages for
some images. However, we have considered some part of the vocabulary, say top
50%, for likelihood generation, as the whole vocabulary is considerably long,
and there are chances of sparsity and negligence of some words due to scarcity
of samples in sentences. To find the correct set of vocabulary, we
experimented to determine the best set and the cutoff. In Figure 2, being a
native speaker of Bengali, we have provided other captions that would be much
more realistic than the translated version.

##  III Learning Network and Representation Description

Learning the representations and correct configuration of the network is
essential for the success of language generators. While image feature is
helpful, semantic distribution features help in enhancing the captions through
direct correlation of the selected objects in images and the likelihood of
words in sentences. In this work, we have introduced new architectures
utilizing different feature spaces and reported their qualitative and
quantitative analysis. While the semantic features are highly sparse, we
introduced a Gaussian smoothing for semantic features and shown that the
caption gets improved with these kinds of smoothing. Smoothing helps in better
capturing of the image region as attention to be decomposed at the hidden
level. In this work, we have focused on compositional and decomposition-based
architectures as they provide some of the state-of-the-art performance [70]
for caption generator without the explicit requirement for large scale image
segmentation and region-based feature extraction like [37]. However, they are
good ways of expansion of the network and generate better descriptions for
images. Our argument is that we can derive better network through
decomposition of the causal representation $\textbf{h}_{t-1}$ than
decomposition of both causal $\textbf{h}_{t-1}$ and previous
$\textbf{x}_{t-1}$ contexts. Sequential recurrent neural network has been
widely used in generative applications and mathematically, the semantic
concept layer is initialized for these networks as the following,

$\textbf{h}_{0}=MLP_{h}(\textbf{v})$ (1) $\textbf{c}_{0}=MLP_{c}(\textbf{v})$
(2)

where v refers to visual features and $MLP(.)$ referred to multiple-layer
perceptron. Figure 1 provided a generalized architectural overview of the
image caption generation system. Later these layer is fused with the Gaussian
smoothen semantic features and provided with the perfect context
representation for each word.

Figure 3: Architectural Comparison of Different LSTM Types: LSTM, GSTag-LSTM
and GSSCN-LSTM.

###  III-A Long Short Term Memory

Long Short Term Memory (LSTM) is widely used sequential recurrent unit because
of its scalability and its unmatched generalized solutions for many sequential
generation related to languages and other distinct representation learning for
classification and most importantly, apart from suppression of the variations
or approximation, like any other deep learning architecture, it helps in
better generalization. LSTM caption generator, with ResNet features, performs
best among all single context processing units for image captioning. The basic
equations for LSTM is provided as the following set of equations.

$\textbf{i}_{t}=\sigma(\textbf{W}_{xi}\textbf{x}_{t-1}+\textbf{W}_{hi}\textbf{h}_{t-1}+\textbf{b}_{i})$
(3)
$\textbf{f}_{t}=\sigma(\textbf{W}_{xf}\textbf{x}_{t-1}+\textbf{W}_{hf}\textbf{h}_{t-1}+\textbf{b}_{f})$
(4)
$\textbf{o}_{t}=\sigma(\textbf{W}_{xo}\textbf{x}_{t-1}+\textbf{W}_{ho}\textbf{h}_{t-1}+\textbf{b}_{o})$
(5)
$\textbf{g}_{t}=\tanh(\textbf{W}_{xg}\textbf{x}_{t-1}+\textbf{W}_{hg}\textbf{h}_{t-1}+\textbf{b}_{g})$
(6)
$\textbf{c}_{t}=\textbf{f}_{t}\odot\textbf{c}_{t-1}+\textbf{i}_{t}\odot\textbf{g}_{t}$
(7) $\textbf{h}_{t}=\textbf{o}_{t}\odot\tanh(\textbf{c}_{t})$ (8)
$\textbf{y}_{t}=\textbf{h}_{t}\textbf{W}_{hy}$ (9)

For categorization, $\textbf{y}_{t}$ is evaluated through convergence to the
categorization distribution through softmax layer defined as,

$\textbf{y}_{t}=\sigma(\textbf{y}_{t})=\frac{\exp(\textbf{y}_{t})}{\sum\limits^{C}_{k=1}\exp((y_{t})_{k})}$
(10)

where we have $\sigma(\textbf{y}_{t})\in\mathbb{R}^{C}\in[0,1]^{C}$ with $C$
as the set of categories. Also we have $\textbf{x}_{t}\in\mathbb{R}^{m}$,
$\textbf{y}_{t}\in\mathbb{R}^{C}$, $\textbf{W}_{hy}\in\mathbb{R}^{C\times d}$,
$\textbf{i},\textbf{f},\textbf{o},\textbf{g},\textbf{c}\in\mathbb{R}^{d}$,
$\textbf{h}_{t}\in\mathbb{R}^{d}$, $\textbf{W}_{x*}\in\mathbb{R}^{d\times m}$,
$\textbf{W}_{h*},\textbf{W}_{c*}\in\mathbb{R}^{d\times d}$,
$\textbf{b}_{*}\in\mathbb{R}^{d}$. The objective minimization function is
defined as $J(\textbf{W})=\arg\min\frac{1}{2s}\sum\limits_{\forall
s}\sum\limits_{i}||y_{t,i}-y_{t,i}^{\prime}||^{2}=\arg\min\frac{1}{2s}\sum\limits_{\forall
s}||\textbf{y}_{t}-\textbf{y}_{t}^{\prime}||^{2}$ and $s$ number of training
samples. The parameters are updated with $\alpha\frac{\partial
J(\textbf{W})}{\partial\textbf{W}}$. Value of $\alpha$ determines the learning
rate for adaption with the changing topology of the objective function space.

$\textbf{x}_{t}=(\arg\max(\textbf{h}_{t}\textbf{W}_{hy}))\textbf{W}_{E}=(\arg\max\textbf{y}_{t})\textbf{W}_{E}$
(11)

where $\textbf{W}_{E}\in\mathbb{R}^{V\times d}$ is the word embedding
representation of vocabulary size $V$ and each word is represented with vector
of dimension $d$. In most of the architecture,
$\textbf{h}_{t}\rightarrow\textbf{x}_{t}$ involves Equation 11 and is a
convenience and may not be explicitly mentioned with every model but is
implied. Extra set of improvement is

Mathematically, Long Short Term Memory (LSTM) model, denoted as
$f_{{}_{L}}(.)$, can be described as the followings probability distribution
estimation.

$\begin{split}f&{}_{{}_{L}}(\textbf{v})=\prod\limits_{k}\mathrm{Pr}(w_{k}\mid\textbf{v},\text{
}\textbf{W}_{L_{1}})\prod\limits\mathrm{Pr}(\textbf{v}\mid I,\text{
}\textbf{W}_{1})\\\
&=\prod\limits_{k}\mathrm{Q}_{IC}(w_{k}\mid\textbf{v})\prod\limits\mathrm{Q}(\textbf{v}\mid
I)\end{split}$ (12)

using the weights of the LSTM in the architecture is denoted as
$\textbf{W}_{L_{1}}$, $w_{i}$ as words of sentences, v as image features,
$\mathrm{Q}_{IC}(.)$ and $\mathrm{Q}(.)$ are the Image Caption and feature
generator function respectively. $\mathrm{Q}(.)$ derives v from image I.

###  III-B Gaussian Smoothened Tag Long Short Term Memory

Gaussian Smoothened Tag Long Short Term Memory (GST-LSTM) involves consecutive
operations of selection and compositional fusion of different context
features, including a semantic layer feature set for better representation.
Unlike GSSCN-LSTM, where the input and hidden layer are decomposed through
fusion, GST-LSTM only involves decomposition of the hidden layer. The hidden
layer gets fade away very quickly, and through decomposition, it gets
regenerated and revived. GST-LSTM involves decomposition of the hidden layer
transformation weights as $\textbf{W}=\textbf{W}_{s}\textbf{s}\textbf{W}_{p}$
where s is directly involved as refined semantic level information and
diagonally sparse for orthogonal selection. Through this architecture, we
demonstrate that despite low weights, it can perform as competitively (or even
better) as GSSCN-LSTM. In principle, a lower number of weights and properly
justified data fusion can help in better inference. GST-LSTM architecture is
marked by the reduction in weights compared to GSSCN-LSTM and performs equally
well as GSSCN-LSTM for image caption generations. For a model with 512
dimensions, GST-LSTM has 20% fewer weights than GSSCN-LSTM, but the reduction
is directly proportional to the dimension of s. GST-LSTM Network with image
features involves the same extra set of equations, where the the initial
states $\textbf{h}_{0}$, $\textbf{c}_{0}$ are derived from Equation 1 and
Equation 2 respectively. However, after each iteration, the hidden state gets
revised by Equation 14 through the selection of sub-region with the tag
features $S$ before the next prediction. The initial state for image memory of
hidden states $\textbf{h}_{t}$, $\textbf{c}_{t}$ get fades away with time, and
extra attention can revive the image quality of the memory and enhance the
performance. The main iteration of the model consists of the following set of
equations.

$\hat{\textbf{S}}=f_{{s}_{m}}(\textbf{S})$ (13)

where we have defined $f_{{s}_{m}}(.)$ as the Gaussian Smoothing function. S
is sparse and operates as a diagonal matrix. In the diagonal matrix, the
columns are orthogonal to each other, and the task of each column is to select
the feature of interest and discard the others. These columns come directly
from the image, and hence we talk about a column tensor and the whole matrix.
As we operate different transformations and train the weights, the strength of
these individual columns degrades or distributed unevenly. Hence, Gaussian
smoothing will help in absorbing better features as it transforms the discrete
points to a better structural absorbent. We call $\hat{\textbf{S}}$ as
Gaussian Smoothen Semantic Features (GSSF).

Figure 4: Explanation of Gaussian Smoothing of Semantic Features

Due to limitations in GPU memory, we have kept the dimension of S and
$\hat{\textbf{S}}$ same.

$\textbf{h}_{t-1}=\textbf{W}_{h,m}\hat{\textbf{S}}\odot\textbf{W}_{h,n}\textbf{h}_{t-1}$
(14)
$\textbf{i}_{t}=\sigma(\textbf{W}_{xi}\textbf{x}_{t-1}+\textbf{W}_{hi}\textbf{h}_{t-1}+\textbf{b}_{i})$
(15)
$\textbf{f}_{t}=\sigma(\textbf{W}_{xf}\textbf{x}_{t-1}+\textbf{W}_{hf}\textbf{h}_{t-1}+\textbf{b}_{f})$
(16)
$\textbf{o}_{t}=\sigma(\textbf{W}_{xo}\textbf{x}_{t-1}+\textbf{W}_{ho}\textbf{h}_{t-1}+\textbf{b}_{o})$
(17)
$\textbf{g}_{t}=\tanh(\textbf{W}_{xg}\textbf{x}_{t-1}+\textbf{W}_{hg}\textbf{h}_{t-1}+\textbf{b}_{g})$
(18)
$\textbf{c}_{t}=\textbf{f}_{t}\odot\textbf{c}_{t-1}+\textbf{i}_{t}\odot\textbf{g}_{t}$
(19) $\textbf{h}_{t}=\textbf{o}_{t}\odot\tanh(\textbf{c}_{t})$ (20)

Figure 5: Overall Architecture of GST-LSTM Caption Generator with Gaussian
Smoothen Semantic Features.

Mathematically, Gaussian Smoothened Tag Long Short Term Memory (GST-LSTM),
denoted as $f_{{}_{GST}}(.)$, can be described as the followings probability
distribution estimation.

$\begin{split}f&{}_{{}_{GST}}(\textbf{v})=\prod\limits_{k}\mathrm{Pr}(w_{k}\mid\textbf{v},\text{
}\hat{\textbf{S}}\odot\textbf{h}_{t-1},\text{
}\textbf{W}_{L_{1}})\prod\limits\mathrm{Pr}(\textbf{S}\mid\textbf{v},\text{
}\textbf{W}_{2})\\\ &\prod\limits\mathrm{Pr}(\textbf{v}\mid I,\text{
}\textbf{W}_{1})\\\
&=\prod\limits_{k}\mathrm{Q}_{IC}(w_{k}\mid\textbf{v},\text{
}\hat{\textbf{S}}\odot\textbf{h}_{t-1})\prod\limits\mathrm{Q}_{S}(\textbf{S}\mid\textbf{v})\prod\limits\mathrm{Q}(\textbf{v}\mid
I)\end{split}$ (21)

using the weights of the LSTM in the architecture is denoted as
$\textbf{W}_{L_{1}}$, $w_{i}$ as words of sentences, v as image features, S as
semantic features from $\mathrm{Q}_{S}(.)$, $\hat{\textbf{S}}$ as Gaussian
Smoothened features, $\mathrm{Q}_{IC}(.)$ and $\mathrm{Q}(.)$ are the Image
Caption and feature generator function respectively. $\mathrm{Q}(.)$ derives v
from image I.

###  III-C Gaussian Smoothened Semantic Long Short Term Memory

Gaussian Smoothened Semantic Long Short Term Memory (GSSCN-LSTM) operated a
combination of decomposition of both previous context and hidden layer
information and compared to GST-LSTM, it involves further decomposition of
features, which are fused to compose representations at a higher level for
caption generation, but at the cost of series of weighted transformation
weights. The set of equations that describe GSSCN-LSTM is provided below.
([42, 70]) used the decomposition techniques in their architecture.

$\hat{\textbf{S}}=f_{{s}_{m}}(\textbf{S})$ (22)

where we have defined $f_{{s}_{m}}(.)$ as the Gaussian Smoothing function.

$\textbf{x}_{*,t}=\textbf{W}_{x,*m}\hat{\textbf{S}}\odot\textbf{W}_{x,*n}\textbf{x}_{t-1}$
(23)
$\textbf{h}_{*,t}=\textbf{W}_{h,*m}\hat{\textbf{S}}\odot\textbf{W}_{h,*n}\textbf{h}_{t-1}$
(24)

where we have S as the tag distribution and $\hat{\textbf{S}}$ is the Gaussian
smoothen tag distribution, v as the image features, $*=i/f/o/g$,
$\textbf{W}_{*m}\in\mathbb{R}^{a\times b}$,
$\textbf{W}_{*n}\in\mathbb{R}^{a\times b}$,
$\textbf{h}_{*,t}\in\mathbb{R}^{a\times b}$ and
$\textbf{x}_{*,t}\in\mathbb{R}^{a\times b}$. The decomposed features are
processed in the memory network through the following set of equations.

$\textbf{i}_{t}=\sigma(\textbf{W}_{xi}\textbf{x}_{i,t}+\textbf{W}_{hi}\textbf{h}_{i,t-1}+\textbf{b}_{i})$
(25)
$\textbf{f}_{t}=\sigma(\textbf{W}_{xf}\textbf{x}_{f,t}+\textbf{W}_{hf}\textbf{h}_{f,t-1}+\textbf{b}_{f})$
(26)
$\textbf{g}_{t}=\sigma(\textbf{W}_{xg}\textbf{x}_{g,t}+\textbf{W}_{hg}\textbf{h}_{g,t-1}+\textbf{b}_{g})$
(27)
$\textbf{o}_{t}=\sigma(\textbf{W}_{xo}\textbf{x}_{o,t}+\textbf{W}_{ho}\textbf{h}_{o,t-1}+\textbf{b}_{o})$
(28)
$\textbf{c}_{t}=\textbf{f}_{t}\odot\textbf{c}_{t-1}+\textbf{i}_{t}\odot\textbf{g}_{t}$
(29) $\textbf{h}_{t}=\textbf{o}_{t}\odot\tanh(\textbf{c}_{t})$ (30)

where $\textbf{W}_{x*}\in\mathbb{R}^{w_{e}\times s}$,
$\textbf{W}_{h*}\in\mathbb{R}^{d\times s}$, where $d$ is the hidden layer
dimension, $w_{e}$ is the word embedding dimension and $s$ is the semantic
dimension. Here, we have replaced semantic feature $\hat{\textbf{S}}$ with
Gaussian Smoothen Semantic Features (GSSF) $\hat{\textbf{S}}$ and a gradual
smoothen feature helps in better selection for the next phase from the
previous states, generated from the model. It must be mentioned that
$\textbf{h}_{t}$ is shadow image representation generated from the image
features and continuous fusion of the semantic features and the image
features.

Mathematically, Gaussian Smoothened Semantic Long Short Term Memory (GSSCN-
LSTM), denoted as $f_{{}_{GSSCN}}(.)$, can be described as the followings
probability distribution estimation.

$\begin{split}f&{}_{{}_{GSSCN}}(\textbf{v})=\prod\limits_{k}\mathrm{Pr}(w_{k}\mid\textbf{v},\text{
}\hat{\textbf{S}}\odot\textbf{h}_{t-1},\text{
}\hat{\textbf{S}}\odot\textbf{x}_{t-1},\text{ }\textbf{W}_{L_{1}})\\\
&\prod\limits\mathrm{Pr}(\textbf{S}\mid\textbf{v},\text{
}\textbf{W}_{2})\prod\limits\mathrm{Pr}(\textbf{v}\mid I,\text{
}\textbf{W}_{1})\\\
&=\prod\limits_{k}\mathrm{Q}_{IC}(w_{k}\mid\textbf{v},\text{
}\hat{\textbf{S}}\odot\textbf{h}_{t-1},\text{
}\hat{\textbf{S}}\odot\textbf{x}_{t-1})\\\
&\prod\limits\mathrm{Q}_{S}(\textbf{S}\mid\textbf{v})\prod\limits\mathrm{Q}(\textbf{v}\mid
I)\end{split}$ (31)

using the weights of the LSTM in the architecture is denoted as
$\textbf{W}_{L_{1}}$, $w_{i}$ as words of sentences, v as image features, S as
semantic features through $\mathrm{Q}_{S}(.)$, $\hat{\textbf{S}}$ as Gaussian
Smoothened features, $\mathrm{Q}_{IC}(.)$ and $\mathrm{Q}(.)$ are the Image
Caption and feature generator function respectively. $\mathrm{Q}(.)$ derives v
from image I.

Figure 6: Overall Architecture of GSSCN-LSTM Caption Generator with Gaussian
Smoothen Semantic Features.

##  IV Details of Methodology

We have provided some implementation level information, though this is very
common for language generator and image captioning research. There are
intricacies of the procedures and are considered as the following.

#### Step 1: Data Acquisition

We have used the MSCOCO data, and some of them are shared by [70], which
coordinates with the ResNet features and can be easily obtained.

#### Step 2: CNN Transfer Learning Feature

Here, we have extensively used ResNet through transfer learning with 2048
dimension.

#### Step 3: Transfer Learning Translation

The Bengali captions are obtained through Google Translation API
(googletrans.Translator) and a plugin provided by them. Since it is a public
plugin and protected by DOS attack, the number of times it can translate for
you on a certain day is limited, and hence it took me several days to
translate the data. Also, since Bengali is unstructured language, the
vocabulary dimension is very large and needs to be cleaned and processed with
by 5% occurrence rate. The English vocabulary is also needed to be shrink-ed
by 5% occurrence rate.

#### Step 4: Transfer Learning Semantic

After the CNN features are extracted, Tag features with 999 dimensions are
used, and this is a neural network transformation of the image features.

#### Step 5: Gaussian Smoothing of Semantic

Here, we explain and introduced Gaussian Smoothen Semantic Features (GSSF).
Figure 4 provided an instance for an illustration of the effects of the
Gaussian Smoothing procedure on semantic features.

#### Step 6: Training Sessions

We trained the LSTM network end-to-end through the use of the all available
data, but without the CNN network as it becomes hefty and prone to over-
fitting.

#### Step 7: Testing in Original Language Space

We processed the evaluation of the generated Bengali sentences with the
ground-truth Bengali sentences.

#### Step 8: Testing in Reference Language Space

For better analysis and clarity, we translated the Bengali sentences and
evaluated the performances in the English language to get an overview in
comparison to the English language caption generator. Also, it provides some
reference of how far we need to go before we can get a perfect captioner. But,
unlike English, which is a grammatically structured language, Bengali (with
90% Sanskrit vocabulary) is an unstructured language, and direct comparison of
relative performance is difficult.

#### Step 9: Qualitative Evaluation for Semantics Correctness

Also, we have made some semantic relatedness through visual inspection in
Qualitative Evaluation section.

##  V Results & Analysis

This section provides some details of the experiments. We have used different
evaluation techniques to put up different aspects of the generated captions in
Bengali language. We also provided instances of the generated caption in
Evaluation of this kind of translation based caption generator analysis is
based on two fronts, mainly because no expert can legitimatize the correctness
of all these languages, and in the absence of grammar, the semantics of these
languages are difficult to define. However, we can offer qualitative
evaluations for the working versions of the sentences in these languages.
Nevertheless, we have provided both qualitative and quantitative metric based
comparisons below.

###  V-A Dataset

We have used the same training-validation-testing split [48] of MSCOCO data
that has been used in all the papers, which consists of 123287
train+validation images and 566747 train+validation sentences. Here, each
image is associated with at least five sentences from a vocabulary of 8791
words. There are 5000 images (with 25010 sentences) for validation and 5000
images (with 25010 sentences) for testing. We used the same data split, as
described in Karpathy’s paper [48]. For our network, two sets of features are
being used: one is ResNet features with 2048 dimension feature vector, and
another is the semantic representation with feature vector of 999 dimensions
and consisted of the likelihood probability of occurrence for the most
appearing set of objects, attributes and interaction criteria as set of
combinations of incidents for the dataset. For the translated data, we have
restricted the vocabulary of the sentences to 20K and used 32 and 64 batch
size based training sessions for each of the models. Also, we found that 1024
dimensions of the hidden layers provided the best performance, while the word
embedding is loaded from Stanford GloVe 300 dimension pre-trained model.

###  V-B Evaluation Metrics

Different metrics like CIDEr-D, METEOR, ROUGE_L, BLEU_n, and SPICE are used
for our evaluation. However, we think that the BLEU_4 metric is the most
sensible way of evaluation as this metric technique takes care of the
continuity of object-attribute relationships and at the same time provides
organizes the best possible descriptions. Also, BLEU_4 evaluation considers
several consecutive lineages based on nouns, adjectives, adverbs from the
ground truth, which can never be judged through other criteria. We used
Microsoft COCO dataset for our evaluation, and the results are reported based
on Karpathy’s division of training, validation, and testing set.

###  V-C Evaluation Procedures

The first evaluation is in Bengal vocabulary space and the performance is
dependent on the vocabulary space of the training model. If we consider the
function of the translation as $Tr(.)$ and the evaluator function as
$f_{Q}(.)$, then we define the two different evaluation criteria as the
followings.

$E_{1}=f_{Q}(\textbf{G}_{L},Tr(\textbf{R}_{E})_{L})$ (32)
$E_{2}=f_{Q}(Tr(\textbf{G}_{L})_{E},\textbf{R}_{E})$ (33)

where $Tr(.)_{L}$ is the translation function to language $L$ and
$\textbf{G}_{L}$ is the generated captions in language $L$ and
$\textbf{R}_{L}$ is the reference caption in language $L$. We define evaluator
function $f_{Q}(.)$ as,

$f_{Q}(.)\in\\{\text{BLEU\\_n, CIDEr, ROUGE\\_L, METEOR, SPICE}\\}$ (34)

Also, The two reference languages frames denoted as $L$ (Any Language other
than English) and $E$ (English). The evaluation criteria had varied ranges of
metrics, and we evaluated the generated sentences with each of them due to the
limitation of grammatical knowledge of different languages and the way the
machine generates captions in different languages. Apart from the different
evaluation metrics, we have also used Beam Search for generating longer
sentences or at least sentences with a higher cumulative probability value.

###  V-D Quantitative Analysis

We made a comparative study of the different architectures and compositional
extraction of different representations. Quantitative analysis helps in
understanding the capability of the model in objects based inference in
sentences and its distance from the ground truth and also their ability to
compose a different form of short sequences from image contexts. Table II
provided a comparative overview of evaluation based on the language itself
based on the reference of that language. In contrast, Table III provided the
same metric evaluation when the generated captions are translated to English
language, and a comparison is made based on the ground truth of the reference
of the English language. From Table II it is evident that the captioner
generator performed very well. Even though the source of data is a translator,
which has not reached perfection and is just a prototype with errors. This
limitation of the translation models is mainly due to the unstructured nature
of the ancient languages like Bengali, which is directly related to Sanskrit
and was there from pre-historic times. From Table III, we can say that our
analysis is limited to constrained vocabulary (20K most appearing Bengali
words) and given a better data, this approaches can be improved with precision
and as a native Bengali speaker, this model can solve the problem of Bengali
language based image captioner applications. We have also used beam search
with beam size as 5 and also made sure that the repetition is prevented from
appearing as it reduces the performance. Also, we have utilized a dropout rate
of 0.5 for the embedding, image features, entry points of the language decoder
and the exit points of the language decoders. 0.5 dropout rate provides ample
scope for all the variables to be selected or rejected entirely and is very
helpful to avoid over-fitting.

###  V-E Discussion

For analysis of the results, we found that our proposed model GST-LSTM model
performed the best and is also much lower (in the number of weights) in
comparison to GSSCN-LSTM and better than LSTM base model in performance. GST-
LSTM out-performed GSSCN-LSTM and LSTM by 1% and 4% respectively for BLEU_4
metric and better data will even improve the relative performances. The main
reason why the GSSF model performed better has diverse explanations. The
addition of the semantic level information and increasing the spread using
Gaussian Smoothening (GSSF) has helped considerably. This approach can improve
the results as it is providing a better concept layer for the generation of
the representation for the captions. GSSF provides a new technique to upgrade
the sparse semantic tensor information, mainly when it is acting as a diagonal
matrix and its prime work is the selection of the structural information from
the media contents. Other factors that helped in better captioning is through
the extraction of the usable vocabulary space and also restricting the word
embedding. We have used the word embedding of Stanford GloVe through the pre-
trained model and gradually smoothened it. Also, for assigning the embedding
vector, we have taken the help of a translator model to map the Bengali words
with its English counterparts. Table II and Table III provided the numerical
results.

TABLE II: Performance Evaluation: Original Generated Vs Original Ground Truth Algorithm | Language | CIDEr-D | Bleu_4 | Bleu_3 | Bleu_2 | Bleu_1 | ROUGE_L
---|---|---|---|---|---|---|---
LSTM | Bengali | 0.93 | 0.31 | 0.42 | 0.55 | 0.70 | 0.53
GST-LSTM | Bengali | 0.99 | 0.35 | 0.48 | 0.63 | 0.77 | 0.57
GSSCN-LSTM | Bengali | 1.01 | 0.34 | 0.44 | 0.58 | 0.73 | 0.55
TABLE III: Performance Evaluation: English Translated of Generated Vs English Ground Truth Algorithm | Language | CIDEr-D | Bleu_4 | Bleu_3 | Bleu_2 | Bleu_1 | ROUGE_L
---|---|---|---|---|---|---|---
LSTM | Bengali | 0.62 | 0.24 | 0.34 | 0.46 | 0.58 | 0.39
GST-LSTM | Bengali | 0.80 | 0.25 | 0.36 | 0.49 | 0.66 | 0.46
GSSCN-LSTM | Bengali | 0.78 | 0.25 | 0.35 | 0.49 | 0.66 | 0.46

###  V-F Qualitative Analysis

It is very difficult to evaluate whether a model is better than the other
through averaged numerical. Hence, we have provided some instances of the
generated captions in Figure 7 and Figure 8. Whether overall improved captions
are generated or not is also difficult to judge from numerical in Table II and
Table III. Hence, the following qualitative analysis is adopted and will
reflect some of them. Most of the caption generated work used diverse
evaluation methods for the generated captions, but there is a requirement of
analysis of quality and context correctness of these languages. Also,
qualitative metrics are the best way of understanding the acceptability of the
generated sentences and hence in this part, we have demonstrated some
instances with the original images. Figure 7 and Figure 8 provided some
instances of the different caption of different Indian languages.

Figure 7: Qualitative Analysis of Captions. Set 1. Figure 8: Qualitative
Analysis of Captions. Set 2.

From these generated captions, we can say that the model performed quite well
and the sentences represent much better description of the situations.

##  VI Conclusion & Future Works

In this work, we have introduced a new memory unit architecture for sequential
learning and utilized it for image captioning applications for different
unstructured (grammatically) Indian languages like Bengali and is the fifth
widely spoken language in the world. The features include a different set of
language attributes and grammatical properties and completely different from
English. While we have used Bengali language as our reference, other Indian
languages can be used in this way as well. The quantitative evaluation
criteria for performance can be regarded as a reference, while the qualitative
evaluation criteria can be seen as a perfect judge for many situation
narrations for images. We devised different architectures (GST-LSTM and GSSCN-
LSTM) to provide a comparison with baseline LSTM and also provided a new
technique to upgrade the sparse semantic tensor information, mainly when it is
acting as a diagonal matrix. Its prime work is a selection of the structural
information from the media contents. Since this is the very first work, lots
of further work can be done in this paradigm and our performance metric can be
used for reference and baseline.

## Acknowledgment

The author has used University of Florida HiperGator, equipped with NVIDIA
Tesla K80 GPU, extensively for the experiments. The author acknowledges
University of Florida Research Computing for providing computational resources
and support that have contributed to the research results reported in this
publication. URL: http://researchcomputing.ufl.edu

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The cross-section for the $\gamma e^{-}\rightarrow Ze^{-}\rightarrow
l^{-}l^{+}e^{-}$ scattering at the LHeC

Bui Thi Ha Gianga<EMAIL_ADDRESS>
a Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam

###### Abstract

A measurement of Z production cross-section in $\gamma e^{-}$ collision at
Large Hadron-electron Collider (LHeC) is presented to compare to that at
International Linear Collider (ILC). The total cross-section depends strongly
on the polarization of the initial and final $e^{-}$ beams, the electron beam
energy $E_{e}$ with the energy of the proton beam taken to be $E_{p}=7$ TeV.
The result shows that the total cross-section in $\gamma e^{-}\rightarrow
Ze^{-}\rightarrow l^{-}l^{+}e^{-}$ at LHeC is much larger than that at ILC.

Keywords: Z production, cross-section, LHeC.

##  I Introduction

The electro-weak and strong interactions have been unified in the Standard
model (SM). The discovery of Higgs signal at the Large Hadron Collider (LHC)
has completed the particle spectrum of the SM. However, the existence of some
theoretical drawbacks in the SM has motivated the extended models. One of the
most attractive extended models is Randall-Sundrum (RS) model which provides a
new scenario with the extra dimension to solve the gauge hierarchy naturally
[1]. Due to the extra dimension, the additional scalar field called radion can
be mixed with Higgs boson [2, 3, 4, 5]. The low mass radion region, even 10
GeV, is worth investigating [6]. Besides the SM couplings, the anomalous
couplings of the mixed radion and Higgs boson to $\gamma\gamma,ZZ,gg,\gamma Z$
have been considered .
It is important to test the electro-weak sector of the SM through studying the
Z boson production. The Z production using the Compton backscattered photon
was considered at International Linear Collider (ILC) [7]. Recently, Z-pair
production in hadron colliders has been investigated at the LHC [8, 9, 10, 11,
12, 13, 14]. However, the Large Hadron-electron Collider (LHeC), planned at
the Large Hadron Collider (LHC), is a cleaner configuration [15, 16, 17, 18,
19]. Moreover, the center of mass energy at the LHeC is higher than that at
the ILC. The incoming proton beam energy at LHeC has been $E_{p}=7$ TeV and
the center of mass energy has been $\sqrt{s}=2\sqrt{E_{p}E_{e}}$. The LHeC has
planned to collide electrons with energy from 60 GeV to 140 GeV [20].
Therefore, it can provide a better condition for researching a lot of new
phenomena compared to the LHC and ILC. The LHeC may play a significant role in
the pursuit of new physics beyond the SM [21]. $\gamma e^{-}$ collision with
the photon beam radiated from the proton has provided an extra experimental
scenario to help reduce the background [22]. In spite of a lower luminosity,
$\gamma e^{-}$ subprocess can be studied as a complementary tool to $e^{-}p$
collision at the LHeC.
In our present work, we have studied the subprocess $\gamma e^{-}\rightarrow
Ze^{-}\rightarrow l^{-}l^{+}e^{-}$, including the vertices of Z boson as,
$\gamma ZZ$, $\gamma\gamma Z$, $\gamma Zh$, $\gamma Z\phi$. With the
contribution of new interactions in RS model, including radion, Higgs
propagators, the total cross-section has been expected to detect in
experiments. The layout of this paper is as follows. In Section II, we review
the Randall-Sundrum model and the mixing of Higgs - radion. The cross-section
for the $\gamma e^{-}\rightarrow Ze^{-}\rightarrow l^{-}l^{+}e^{-}$ collision
at the LHeC is presented in Section III. Finally, we summarize our results and
make conclusions in Section IV.

##  II A review of Randall-Sundrum model and the mixing of Higgs-radion

The RS model consists of one extra dimension bounded by two 3-branes citeahm.
The UV-brane, which the fifth dimension is bounded at, located at $y=0$ and
the IR-brane located at $y=\pi r_{c}$. The five dimensional metric has the
following form:

$ds^{2}=e^{-2kb_{0}|y|}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-b_{0}^{2}dy^{2},$ (1)

where $b_{0}$ is a length parameter for the fifth dimension, $r_{c}$ is the
compactification radius and $k$ is the curvature of the five dimensional
geometry. The exponential represents the warp factor which generates the gauge
hierarchy. The values of the bare parameters are determined by the Planck
scale and the applicable value for size of the extra dimension is assessed by
$kr_{c}\simeq 11-12$. Thus the weak and the gravity scales can be naturally
generated. The relation $\overline{M}_{Pl}^{2}=M_{5}^{3}/k$ is derived from
the 5D action. The scale of physical phenomena on the IR-brane is given by
$\Lambda_{\phi}\equiv\overline{M}_{Pl}e^{-kr_{c}\pi}$ with
$\Lambda_{\phi}\sim$ few TeV. The mass of the $n^{th}$ KK graviton excitation
is $m^{G}_{n}=x_{n}^{G}k\Lambda_{\phi}/\overline{M}_{Pl}$, with $x_{n}^{G}$
being the roots of the $J_{1}$ Bessel function [23]. The coupling strength of
the graviton KK states to the SM fields is $\Lambda_{\phi}^{-1}$. The gravity-
matter interactions have been obtained [24]

$\mathcal{L}_{int}=-\dfrac{1}{\hat{\Lambda}_{W}}\Sigma_{n\neq
0}h^{n}_{\mu\nu}T^{\mu\nu}-\dfrac{\phi_{0}}{\Lambda_{\phi}}T^{\mu}_{\mu},$ (2)

where $h^{n}_{\mu\nu}(x)$ are the Kaluza-Klein (KK) modes of the graviton
field $h_{\mu\nu}(x,y)$, $\phi_{0}(x)$ is the radion field,
$\Lambda_{\phi}\equiv\sqrt{6}M_{Pl}\Omega_{0}$ is the VEV of the radion field
and $\hat{\Lambda}_{W}\equiv\sqrt{2}M_{Pl}\Omega_{0}$. The $T^{\mu\nu}$ is the
energy-momentum tensor, which is given at the tree level [25, 26]

$T^{\mu}_{\mu}=\Sigma_{f}m_{f}\overline{f}f-2m^{2}_{W}W^{+}_{\mu}W^{-\mu}-m^{2}_{Z}Z_{\mu}Z^{\mu}+(2m^{2}_{h_{0}}h_{0}^{2}-\partial_{\mu}h_{0}\partial^{\mu}h_{0})+...$
(3)

The gravity-scalar mixing is described by the following action[27]

$S_{\xi}=\xi\int d^{4}x\sqrt{g_{vis}}R(g_{vis})\hat{H}^{+}\hat{H},$ (4)

where $\xi$ is the mixing parameter, $R(g_{vis})$ is the Ricci scalar for the
metric $g_{vis}^{\mu\nu}=\Omega_{b}^{2}(x)(\eta^{\mu\nu}+\varepsilon
h^{\mu\nu})$ induced on the visible brane,
$\Omega_{b}(x)=e^{-kr_{c}\pi}(1+\frac{\phi_{0}}{\Lambda_{\phi}})$ is the warp
factor, $\hat{H}$ is the Higgs field in the 5D context before rescaling to
canonical normalization on the brane. With $\xi\neq 0$, there is neither a
pure Higgs boson nor pure radion mass eigenstate. This $\xi$ term mixes the
$h_{0}$ and $\phi_{0}$ into the mass eigenstates $h$ and $\phi$ as given by

$\left(\begin{array}[]{c}{h_{0}}\\\
{\phi_{0}}\end{array}\right)=\left(\begin{array}[]{cc}{1}&{6\xi\gamma/Z}\\\
{0}&{-1/Z}\end{array}\right)\left(\begin{array}[]{cc}{\cos\theta}&{\sin\theta}\\\
{-\sin\theta}&{\cos\theta}\end{array}\right)\left(\begin{array}[]{c}{h}\\\
{\phi}\end{array}\right)=\left(\begin{array}[]{cc}{d}&{c}\\\
{b}&{a}\end{array}\right)\left(\begin{array}[]{c}{h}\\\
{\phi}\end{array}\right),$ (5)

where
$Z^{2}=1+6\gamma^{2}\xi\left(1-\,\,6\xi\right)=\beta-36\xi^{2}\gamma^{2}$ is
the coefficient of the radion kinetic term after undoing the kinetic mixing,
$\gamma=\upsilon/\Lambda_{\phi},\upsilon=246$ GeV,
$a=-\dfrac{cos\theta}{Z},b=\dfrac{sin\theta}{Z},c=sin\theta+\dfrac{6\xi\gamma}{Z}cos\theta,d=cos\theta-\dfrac{6\xi\gamma}{Z}sin\theta$.
The mixing angle $\theta$ is

$\tan 2{\theta}=12{\gamma\xi
Z}\frac{m_{h_{0}}^{2}}{m_{\phi_{0}}^{2}-m_{h_{0}}^{2}\left(Z^{2}-36\xi^{2}\gamma^{2}\right)},$
(6)

where $m_{h_{0}}$ and $m_{\phi_{0}}$ are the Higgs and radion masses before
mixing.
The new physical fields h and $\phi$ in (5) are Higgs-dominated state and
radion, respectively

$m_{h,\phi}^{2}=\frac{1}{2Z^{2}}\left[m_{\phi_{0}}^{2}+\beta
m_{h_{0}}^{2}\pm\sqrt{(m_{\phi_{0}}^{2}+\beta
m_{h_{0}}^{2})^{2}-4Z^{2}m_{\phi_{0}}^{2}m_{h_{0}}^{2}}\right].$ (7)

There are four independent parameters
$\Lambda_{\phi},\,\,m_{h},\,\,m_{\phi},\,\,\xi$ that must be specified to fix
the state mixing parameters. We consider the case of $\Lambda_{\phi}=5$ TeV
and $\frac{m_{0}}{M_{P}}=0.1$, which makes the radion stabilization model most
natural [26].

##  III The cross-section for the $\gamma e^{-}\rightarrow Ze^{-}\rightarrow
l^{-}l^{+}e^{-}$ collision

Feynman rules for the couplings in the RS model are shown as follows [6]

$\displaystyle
g_{eeh}=-i\overline{g}_{eeh}=-i\dfrac{gm_{e}}{2m_{W}}\left(d+\gamma b\right),$
(8) $\displaystyle
g_{ee\phi}=-i\overline{g}_{ee\phi}=-i\dfrac{gm_{e}}{2m_{W}}\left(c+\gamma
a\right),$ (9) $\displaystyle C_{\gamma
Zh}=\dfrac{\alpha}{2\pi\nu_{0}}\left[2g_{h}^{r}\left(\dfrac{b_{2}}{tan\theta_{W}}-b_{Y}tan\theta_{W}\right)-g_{h}\left(A_{F}+A_{W}\right)\right],$
(10) $\displaystyle C_{\gamma
Z\phi}=\dfrac{\alpha}{2\pi\nu_{0}}\left[2g_{\phi}^{r}\left(\dfrac{b_{2}}{tan\theta_{W}}-b_{Y}tan\theta_{W}\right)-g_{\phi}\left(A_{F}+A_{W}\right)\right].$
(11)

where a, b, c, d are the state mixing parameters in RS model. $g_{h}=d+\gamma
b$, $g_{\phi}=c+\gamma a$, $g_{h}^{r}=\gamma b$, $g_{\phi}^{r}=\gamma a$, the
triangle loop functions $A_{F},A_{W}$ are given in Ref.[28]
We consider the collision process in which the initial state contains electron
and photon which emitted from proton beams, the final state contains Z boson
and electron

$e^{-}(p_{1})+\gamma(k_{2})\rightarrow e^{-}(k_{1})+Z(k_{2}).$ (12)

Here, $p_{i},k_{i}$ (i = 1, 2) stand for the momentums. There are three
Feynman diagrams contributing to reaction (12), representing the s, u,
t-channel exchange depicted in Fig.1.

Figure 1:  Feynman diagrams for $\gamma e^{-}\rightarrow Ze^{-}\rightarrow
l^{-}l^{+}e^{-}$ collision, representing the s, u, t-channels, respectively.

The transition amplitude representing the s-channel is given by

$M_{s}=-i\dfrac{e\overline{g}_{eZ}}{q^{2}_{s}-m^{2}_{e}}\varepsilon^{*}_{\nu}(k_{2})\gamma^{\nu}\left(v_{e}-a_{e}\gamma^{5}\right)\overline{u}(k_{1})\left(\not{q}_{s}+m_{e}\right)\varepsilon_{\mu}(p_{2})\gamma^{\mu}u(p_{1}).$
(13)

The transition amplitude representing the u-channel can be written as

$M_{u}=-i\dfrac{e\overline{g}_{eZ}}{q^{2}_{u}-m^{2}_{e}}\overline{u}(k_{1})\gamma^{\mu}\varepsilon_{\mu}(p_{2})\left(\not{q}_{u}+m_{e}\right)\varepsilon^{*}_{\nu}(k_{2})\gamma^{\nu}\left(v_{e}-a_{e}\gamma^{5}\right)u(p_{1}).$
(14)

The transition amplitude representing the t-channel is given by

$M_{t}=M_{Z}+M_{\gamma}+M_{\phi}+M_{h},$ (15)

here

$\displaystyle
M_{\gamma}=-\dfrac{e}{q^{2}_{t}}\overline{u}(k_{1})\gamma^{\beta}u(p_{1})\eta_{\sigma\beta}\varepsilon^{*}_{\nu}(k_{2})\Gamma^{\mu\sigma\nu}_{\gamma\gamma
Z}(p_{2}q_{t}k_{2})\varepsilon_{\mu}(p_{2}),$ (16) $\displaystyle
M_{Z}=-\dfrac{\overline{g}_{eZ}}{q^{2}_{t}-m^{2}_{Z}}\overline{u}(k_{1})\gamma^{\beta}\left(v_{e}-a_{e}\gamma^{5}\right)u(p_{1})\left(\eta_{\sigma\beta}-\dfrac{q_{t\sigma}q_{t\beta}}{m^{2}_{Z}}\right)\varepsilon^{*}_{\nu}(k_{2})\Gamma^{\mu\sigma\nu}_{\gamma
ZZ}(p_{2}q_{t}k_{2})\varepsilon_{\mu}(p_{2}),$ (17) $\displaystyle
M_{\phi}=i\dfrac{\overline{g}_{ee\phi}C_{\gamma
Z\phi}}{q^{2}_{t}-m^{2}_{\phi}}\overline{u}(k_{1})u(p_{1})\varepsilon^{*}_{\nu}(k_{2})\left(p_{2}k_{2}\eta^{\mu\nu}-p_{2}^{\nu}k_{2}^{\mu}\right)\varepsilon_{\mu}(p_{2}),$
(18) $\displaystyle M_{h}=i\dfrac{\overline{g}_{eeh}C_{\gamma
Zh}}{q^{2}_{t}-m^{2}_{h}}\overline{u}(k_{1})u(p_{1})\varepsilon^{*}_{\nu}(k_{2})\left(p_{2}k_{2}\eta^{\mu\nu}-p_{2}^{\nu}k_{2}^{\mu}\right)\varepsilon_{\mu}(p_{2}).$
(19)

The triple gauge boson couplings are given by [14]

$\Gamma^{\sigma\mu\nu}_{\gamma\gamma
Z}(p_{2}q_{t}k_{2})=\dfrac{g_{e}}{m_{Z}^{2}}p_{2}p_{2}\Biggl{[}h_{1}^{\gamma}(q_{t}^{\mu}\eta^{\sigma\nu}-q_{t}^{\nu}\eta^{\sigma\mu})-h_{3}^{\gamma}\varepsilon^{\mu\sigma\nu\alpha}q_{t\alpha}\Biggr{]},$
(20) $\Gamma^{\sigma\mu\nu}_{\gamma
ZZ}(p_{2}q_{t}k_{2})=-\dfrac{g_{e}}{m_{Z}^{2}}p_{2}p_{2}\Biggl{[}f_{4}^{\gamma}(p_{2}^{\nu}\eta^{\mu\sigma}+p_{2}^{\sigma}\eta^{\mu\nu})-f_{5}^{\gamma}\varepsilon^{\mu\sigma\nu\alpha}(q_{t\alpha}-k_{2\alpha})\Biggr{]},$
(21)

The total cross-section for the whole process can be calculated as follows

$\sigma=\sigma(\gamma e^{-}\rightarrow Ze^{-})\times Br(Z\rightarrow
l^{-}l^{+}).$ (22)

The effective production cross-section for the subprocess at the LHeC can be
described [22]

$\sigma(\gamma e^{-}\rightarrow
Ze^{-})=\int^{\xi_{max}}_{Max(\dfrac{(m_{e}+m_{Z})^{2}}{s},\xi_{min)}}E_{p}f(\xi
E_{p})d\xi\int^{(cos\psi)_{max}}_{(cos\psi)_{min}}\dfrac{d\widehat{\sigma}(\widehat{s})}{dcos\psi}dcos\psi,$
(23)

where

$\frac{d{\widehat{\sigma}(\widehat{s})}}{d(cos\psi)}=\frac{1}{32\pi\widehat{s}}\frac{|\overrightarrow{k}_{1}|}{|\overrightarrow{p}_{1}|}|M_{fi}|^{2}$
(24)

is the expressions of the differential cross-section [29],
$\widehat{s}=4E_{e}E_{\gamma}=\xi s$ is center of mass energy of the sub-
process $\gamma e^{-}\rightarrow Ze^{-}$.
$\psi=(\widehat{\overrightarrow{p}_{1},\overrightarrow{k}_{2}})$ is the
scattering angle. The photon flux can be written as

$f(\xi
E_{p})=\int_{Q^{2}_{min}}^{Q^{2}_{max}}\dfrac{dN_{\gamma}}{dE_{\gamma}dQ^{2}}dQ^{2},$
(25)

where

$\dfrac{dN_{\gamma}}{dE_{\gamma}dQ^{2}}=\dfrac{\alpha_{e}}{\pi}\dfrac{1}{E_{\gamma}Q^{2}}\left[\left(1-\dfrac{E_{\gamma}}{E_{p}}\right)\left(1-\dfrac{Q^{2}_{min}}{Q^{2}}\right)F_{E}+\dfrac{E^{2}_{\gamma}}{2E^{2}_{p}}F_{M}\right],$
(26)

with

$\displaystyle
Q^{2}_{min}=\dfrac{M^{2}_{P}E^{2}_{\gamma}}{E_{p}(E_{p}-E_{\gamma})},$ (27)
$\displaystyle
F_{E}=\dfrac{4M^{2}_{P}G^{2}_{E}+Q^{2}G_{M}^{2}}{4M^{2}_{P}+Q^{2}},$ (28)
$\displaystyle
G^{2}_{E}=\dfrac{G^{2}_{M}}{\mu^{2}_{P}}=\left(1+\dfrac{Q^{2}}{Q^{2}_{0}}\right)^{-4},$
(29) $\displaystyle F_{M}=G^{2}_{M}.$ (30)

$M_{P}$ is the mass of the proton. $E_{p}$ is the energy of the incoming
proton beam. $E_{\gamma}=\xi E_{p}$ is the photon energy, which is related to
the loss energy of the emitted proton beam. $F_{E},F_{M}$ are functions of the
electric and magnetic form factors given in the dipole approximation.
$\mu^{2}_{P}=7.78$ is the magnetic moment of the proton.
For numerical calculations, we choose the energy of the incoming proton beam
at LHeC $E_{p}=7$ TeV, the electron beam energy $E_{e}$ in the range 60 GeV
$\leq E_{e}\leq 140$ GeV [20]. The vacuum expectation value (VEV) of the
radion field is $\Lambda_{\phi}=5$ (TeV) [27]. The radion mass has been
selected $m_{\phi}=10$ GeV [30]. The Higgs mass $m_{h}=125$ GeV (CMS). The
maximum value of anomalous couplings in the tightest limits with the
corresponding observable are chosen as $f_{4}^{\gamma}=2.4\times 10^{-3}$,
$f_{5}^{\gamma}=2.7\times 10^{-3}$, $h_{1}^{\gamma}=3.6\times 10^{-3}$,
$h_{3}^{\gamma}=1.3\times 10^{-3}$ [14]. The polarization coefficients of
initial and final $e^{-}$ beams are taken to be $P_{1}=P_{2}=0.8$,
respectively [17]. Since the contribution to the above integral formula is
very small for $Q^{2}_{max}>2GeV^{2}$, therefore $Q^{2}_{max}$ is chosen as 2
$GeV^{2}$ [22]. We give estimates for the cross-sections as follows
i) The electron beam energy is chosen as $E_{e}=60$ GeV. In Fig.2, we plot the
differential cross-section as a function of the $cos\psi$. From the figure, we
can see that $d\sigma/dcos\psi$ is peaked in the forward direction and flat in
the backward direction.
ii) In Fig.3, the forward-backward asymmetry is plotted as a function of
$E_{e}$. The figure indicates that the forward-backward asymmetry decreases
when the electron beam energy increases.
iii) In Fig.4, we evaluate the dependence of the total cross-section on the
initial and final electron beam in the case of $E_{e}=60$ GeV. The cross-
section achieves the maximum value when both initial and final electron beams
are left or right polarized $(P_{1}=P_{2}=\pm 1)$. The cross-section has the
minimum value when $P_{1}=1,P_{2}=-1$ and vice versa. It is worth noting that
the cross-section in the case of both left or right polarized initial and
final electron beams is twice as much as the unpolarized electron beams.
iv) In Fig.5, we evaluate the dependence of the total cross-section on the
electron beam energy $E_{e}$. The electron beam energy is chosen in the range
of 60 GeV$\leq\sqrt{s}\leq$ 140 GeV. Here, the polarization coefficients of
initial and final electron beam are chosen as $P_{1}=P_{2}=0.8,0.6,0$. The
figure shows that the total cross-sections increase when the electron beam
energy $\sqrt{s}$ increases. In Table 1, some typical values for the total
cross-section are measured. We emphasize that the main contribution to the
description process is by the s, u-channels, while the anomalous couplings for
the contribution are very small. However, the study of anomalous couplings
concerning the mixing of Higgs-radion needs to be further studied.

Figure 2:  The differential cross-section with respect to the $cos\psi$. The parameters are chosen as $P_{1}=P_{2}=0.8$, $E_{e}$ = 60 GeV.  Figure 3:  The forward-backward asymmetry with respect to the electron beam energy $E_{e}$. The parameters are taken to be $P_{1}=P_{2}=0.8$.  Figure 4:  The total cross-section as a function of the polarization coefficients of initial and final electron beam. The electron beam energy is chosen as $E_{e}=60$ GeV.  Figure 5:  The total cross-sections as a function of the electron beam energy $E_{e}$ in the cases of the polarization coefficients $P_{1}=P_{2}=0.8,0.6,0$. Table 1: Some typical values for the total cross-section in the $\gamma e^{-}\rightarrow Ze^{-}\rightarrow l^{-}l^{+}e^{-}$ collisions at the LHeC. $E_{e}$ (GeV) | 60 | 70 | 80 | 90 | 100 | 110 | 120 | 130 | 140
---|---|---|---|---|---|---|---|---|---
$\sigma$($P_{1}=P_{2}=0.8$) ($10^{3}pb$) | 0.069 | 0.098 | 0.133 | 0.174 | 0.221 | 0.274 | 0.333 | 0.399 | 0.471
$\sigma$($P_{1}=P_{2}=0.6$) ($10^{3}pb$) | 0.057 | 0.081 | 0.110 | 0.144 | 0.183 | 0.227 | 0.276 | 0.331 | 0.390
$\sigma$($P_{1}=P_{2}=0$) ($10^{3}pb$) | 0.042 | 0.059 | 0.081 | 0.106 | 0.135 | 0.167 | 0.203 | 0.243 | 0.287

##  IV Conclusion

In this work, we have evaluated the total cross-section in $\gamma
e^{-}\rightarrow Ze^{-}\rightarrow l^{-}l^{+}e^{-}$ production at LHeC in the
RS model. The result shows that the main contribution to the description
process is by the s, u-channels, while the anomalous couplings for the
contribution are very small. Note that the Z boson with such a short lifetime
is never seen directly in the experiments. Even though the branching ratio of
$Z\rightarrow l^{-}l^{+}$ is fairly small, the $l^{-}l^{+}$ channel leaves a
relatively clean signature in the detector, which allows the process to be
measured with a higher accuracy [31]. In this study, our interest is
comparison between the cross-section at ILC and at LHeC. The value at ILC has
been read in detail in Ref.[7]. The results that the total cross-section in
$\gamma e^{-}\rightarrow Ze^{-}\rightarrow l^{-}l^{+}e^{-}$ at LHeC is much
larger than that at ILC.
Finally, we emphasize that the scalar anomalous concerning the mixing of
Higgs- radion in the RS model needs to be investigated further.
Acknowledgements: The work is supported in part by Hanoi National University
of Education under Grant No. SPHN21 – 07.

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