# Utilizing Large Language Models to Identify Reddit Users Considering Vaping
Cessation for Digital Interventions

Sai Krishna Revanth Vuruma1, Dezhi Wu1,4, Saborny Sen Gupta1, Lucas Aust1,
Valerie Lookingbill1,
Caleb Henry1, Yang Ren1, Erin Kasson2, Li-Shiun Chen2, Patricia Cavazos-Rehg2,
Dian Hu3, Ming Huang3,4

1 HI3 Tech Lab
Department of Integrated Information Technology
University of South Carolina
Columbia, SC, USA
2 School of Medicine
Washington University
St. Louis, MO, USA
3 Center of Translational AI Excellence and Applications in Medicine
McWilliams School of Biomedical Informatics
University of Texas Health Science Center
Houston, TX, USA
4Corresponding Authors: Dezhi Wu<EMAIL_ADDRESS>Ming Huang,
<EMAIL_ADDRESS>
###### Abstract

The widespread adoption of social media platforms globally not only enhances
users’ connectivity and communication but also emerges as a vital channel for
the dissemination of health-related information, thereby establishing social
media data as an invaluable organic data resource for public health research.
The surge in popularity of vaping or e-cigarette use in the United States and
other countries has caused an outbreak of e-cigarette and vaping use-
associated lung injury (EVALI), leading to hospitalizations and fatalities in
2019, highlighting the urgency to comprehend vaping behaviors and develop
effective strategies for cession. In this study, we extracted a sample dataset
from one vaping sub-community on Reddit to analyze users’ quit vaping
intentions. Leveraging large language models including both the latest GPT-4
and traditional BERT-based language models for sentence-level quit-vaping
intention prediction tasks, this study compares the outcomes of these models
against human annotations. Notably, when compared to human evaluators, GPT-4
model demonstrates superior consistency in adhering to annotation guidelines
and processes, showcasing advanced capabilities to detect nuanced user quit-
vaping intentions that human evaluators might overlook. These preliminary
findings emphasize the potential of GPT-4 in enhancing the accuracy and
reliability of social media data analysis, especially in identifying subtle
users’ intentions that may elude human detection.

###### Index Terms:

Vaping Cessation, Machine Learning, Large Language Models, Social Media
Analytics, Natural Language Processing, GPT-4 Annotation, BERT Classification,
Reddit

##  I Introduction

Due to the ubiquity of social media platforms, over 4.7 billion users
worldwide use them for connectivity, communications, news, and entertainment
with a significant portion of the discourse related to health. The utilization
of social media data emerges as a nascent source of public health information,
offering novel insights into public health trends and enhancing the
capabilities of public health surveillance.

In recent years, the United States has witnessed a significant surge in the
popularity of vaping or e-cigarette use, leading to a notable rise in cases of
e-cigarette and vaping use-associated lung injury (EVALI) that caused
hospitalizations and fatalities in 2019. Previous research studies have
leveraged popular Twitter and Reddit social media data for public surveillance
of health topics. In particular, two previous social media vaping studies [1,
2] used topic modeling and sentiment analysis along with clinical insights to
show social media users might benefit from digital intervention programs for
vaping cessation and potentially for clinicians to employ proactive outreach
strategies to engage vaping patients for patient education and treatment on
social media platforms. Moreover, in some other recent studies on e-cigarette
or vape use, many users reported symptoms of Vape Dependence[3], where they
cannot go for long periods of time with out craving for a vape, with users
commonly resorting to stealth-vaping[4] in places that prohibit vaping.
Further, researchers found that vape frequency was directly associated with
perceived satisfaction while being indirectly associated with perceived
danger[5]. Alarmingly, a teenager vaping study [6] has highlighted the youth
as high risk population for targeted health communications. To tackle this
pressing public health challenges, it is imperative to conduct further
research into the analysis of vaping discourse on social media platforms,
aiming to develop artificial intelligent (AI)-based approach to more
efficiently and accurately identify these patients’ social media vaping
discourse behaviors and developing targeted vaping prevention and intervention
programs for the youth population.

In this study, we aim to employ and evaluate OpenAI’s GPT-4 model and
traditional BERT-based language models and by comparing human evaluators on a
data annotation task to identify vaping cession interests among Reddit users.
Our preliminary findings indicate GPT-4 model demonstrates better consistency
in adhering to annotation guidelines and processes compared to human
evaluators. Following the introduction, in the next section, we present the
related work in this area, and then report our research methodology and
preliminary study findings. Lastly, we draw study conclusion and propose
future research directions.

##  II Related Work

In this technologically prudent era, social media platforms, such as Facebook,
Instagram, Reddit, and Twitter, have emerged as pivotal spaces for self-
expression and social interaction [7]. These platforms offer a massive data
resource that reflects the perceptions of its users, and the extraction and
analysis of these data extends an opportunity for different research studies
[7, 8]. However, interpretation of natural language data requires deep
contextual knowledge and understanding [9].

Manual annotations of social media texts can be challenging for humans as the
texts are short and informal and contain different socio-cultural opinions and
perceptions [7, 10]. Studies have shown that lack of contextual understanding
can lead to incorrect annotations [10]. Recent work has shown that data
annotation process can be augmented with state-of-the-art Machine Learning
(ML) algorithms and Natural Language Processing (NLP) models to great success
[11]. However, care must be taken while using these techniques in tasks
requiring complex inferences as shown in [9, 12]. In this regard, advanced and
intuitive Large Language Models (LLMs) such as OpenAI’s Generative Pre-trained
Transformer models GPT-3[13] and GPT-4[14] and Meta’s LLAMA 2 [15] have gained
popularity in recent years due to their proficiency in in-context learning
where they outperformed traditional methods [16, 17]. These models can
generate quick results and are not susceptible to some of the limitations seen
in human annotation [18].

Using the GPT models, the findings from [19] show that while GPT-3 still has
room for improvement when compared to human annotators, it is capable of
annotating data from different domains. Another study used GPT-4 to label the
political tweets collected from USA politicians [9] with better accuracy,
reliability, and equal or lower bias of GPT-4 than the expert classifiers.

##  III Methodology

Reddit is a free, popular social media platform where users can take part in
discussions and share content. It facilitates conversations between like-
minded users through dedicated communities called subreddits. Each subreddit
is focused on a specific topic or interest shared amongst it’s subscribers.

r/QuitVaping is a subreddit where users come together to motivate each other
to quit vaping. With around 40,000 subscribers, it is the largest subreddit
dedicated to help users quit vaping and other tobacco products.

The workflow adopted for this study is illustrated in Figure 1. Each stage of
the pipeline will be discussed in detail in the respective subsections. First
the data is extracted and cleaned from Reddit, then sent to the human
evaluators and the GPT-4 model for annotations. The human annotated dataset is
then passed to pre-trained BERT models for training on a classification task.
The performance of all three approaches is compared at the end to draw
conclusions.

Figure 1: Workflow to evaluate GPT-4 and BERT-based language models

###  III-A Data Collection & Preparation

Reddit’s Async PRAW API provides direct access to posts by subreddit through
asynchronous execution. From the aforementioned r/QuitVaping subreddit, a
total of 1000 posts made by users were extracted using the API. These posts
ranged from users talking about their progress towards quitting vaping to
users looking for help or motivation to quit or reduce vape use.

Out of these 1000 posts, approximately 120 of them were randomly selected to
form a sample dataset. From each post in the sample dataset, two columns
including title and body were extracted and broken down into sentences using
the Sentence Tokenizer from the NLTK library [20]. Table I gives a statistical
overview of the text in the title and body fields of each post in the sample
dataset.

TABLE I: ANALYSIS OF SENTENCES AND WORDS PER REDDIT POST Field Name |  Average number of sentences per post |  Average number of words per post
---|---|---
Title |  1.07 |  6.56
Body |  9.02 |  157.74

###  III-B Human Annotation

The vaping-related sentences from the extracted sample were assigned to two
human evaluators for manual annotations related quit-vaping intention. The
assigned labels from this step will serve as the ground truth while making
performance comparisons among the ML approaches.

The annotators were tasked to label each sentence as ’Yes’ or ’No’ by
following these guidelines to differentiate whether it is a quit-vaping
related sentence:

  * •

Yes:

    * –

User explicitly mentions their desire, plan, or decision to specifically quit
vaping, e.g.,  ”I have been wanting to quit the vape for a while”

  * •

No:

    * –

Sentences that do not mention vaping or quitting, e.g., ”Sorry if it sounds
overdramatic.”

    * –

User explicitly states that they have already quit vaping, e.g., ”5 days
strong of cold turkey quitting all nicotine.”

    * –

User describes a negative outcome of vaping, such as a negative health
outcome, e.g., ”It makes my mouth taste and smell weird.”

    * –

User expresses negative feelings of or experiences with vaping, such as
feelings of shame or embarrassment about vaping, e.g., ”I work in healthcare
and live a healthy lifestyle and am ashamed of vaping.”

The class distribution after human annotation was concluded can be found in
Table II.

TABLE II: Class Distribution after Human Annotation Quit Vaping Intention |  Training Data |  Validation Data |  Testing Data | Total
---|---|---|---|---
Yes |  64 |  9 |  13 | 86
No |  860 |  114 |  173 | 1147
Total |  924 |  123 |  186 | 1233

###  III-C Classification using pre-trained BERT models

BERT[21] has revolutionized language representations by introducing the
concept of bidirectional attention in transformers. It served as the precursor
to modern day Large Language Models (LLMs). Due to its architecture it can
easily be fine-tuned to perform a multitude of tasks from question answering
to natural language understanding. It can also be adopted to more specific use
cases in social media studies such as COVID19 content analysis[22], Fake News
Detection[23] or Hate Speech Detection[24] making it an ideal candidate for
our purpose.

In order to fine-tune the base BERT model on our data, the sample dataset
after human annotation was passed to the model as the input. The model was
trained to classify the sentences from the title and body columns of the posts
into ’Yes’ or ’No’ instances using the labels assigned by human evaluators as
the target. The labelled dataset was broken down into train (75%), validation
(10%) and test (15%) sets using stratified sampling. Given the imbalance
between the classes in the training data, we had to adjust the model weights
to make if a fair classification. As shown in Table II, the samples with label
’No’ outnumber those with ’Yes’ by about 13.5 times. To account for this
lopsidedness, we assigned class weights for both classes and updated the loss
function during training and evaluation to arrive at a more representative
loss value.

Along with the base model, other pre-trained BERT models such as BioBERT[25],
RedditBERT and DistilBERT[26] were also trained on the data using the same
approach.

###  III-D GPT-4 Annotation

In addition to using traditional transformer models, OpenAI’s GPT-4 model was
used to annotate the data. It is the latest model in the line of the popular
Generative Pre-trained Transformers from OpenAI. It has shown remarkable
capabilities in a multitude of domains, even clearing the bar exam according
to a recent study[27]. Recent studies indicate that GPT-4 can act as an
alternative to human annotation in many diverse areas[9, 12, 28].

To test GPT-4’s ability to annotate Reddit data from our sample dataset, we
used OpenAI’s API feature that lets users interact with the model of their
choice via prompts. The model was tasked to do the following:

  * •

Follow the same guidelines that the human evaluators did for data annotation.

  * •

Assign the label ’Yes’ to a sentence if it indicates the speaker’s intention
to quit vaping right now or the label ’No’ if it does not.

  * •

Identify specific phrases from each sentence to justify your response.

  * •

Provide an explanation of why that phrase does or does not indicate an
intention to quit vaping.

##  IV Results

###  IV-A Pre-trained BERT language models

Four pre-trained variants of the BERT model were fine-tuned on a binary
classification task using the labels from human annotation as the ground
truth. The summary of the classification results on the test dataset for each
model is shown in Table III. Although the overall accuracy and recall suggest
that the models are able to generalize predictions on unseen data, all four
variants had issues identifying the positive class (’Yes’ instances) on the
test dataset. This can in part be owed to the fewer number of ’Yes’ cases in
the training data, even after the loss function was adjusted to better reflect
the train dataset.

Out of the four BERT variants that were tested, BioBERT had the best
classification metrics on the test dataset with an F1 score of 95. On the
’Yes’ instances in the test data, it had the best recall of about 60%. In
comparison, the variant with the second best performance had a recall of 43%
on the positive instances.

DistilBERT on the other hand, had the lowest recall (0%) for the positive
class on the test dataset. It was unable to correctly classify any instance of
the positive class. The base BERT model and RedditBERT were able to make
inferences closer to the BioBERT model. However, their performance on the
’Yes’ instances was not up to par with BioBERT.

TABLE III: BERT Models’ Performance on Classification Model | Accuracy | Recall | F1 Score
---|---|---|---
bert-base-uncased | 94% | 95% | 94
distilbert-base-uncased | 85% | 92% | 89
reddit-bert-text_10 | 93% | 94% | 93
biobert-base-cased-v1.2 | 95% | 96% | 95

###  IV-B GPT-4

While interacting with the API, two fields including role and prompt inform
the model what its intended purpose is. ’Role’ refers to the role of the model
in this conversation while the ’prompt’ field is used to provide input text to
the model. We observed differences in model performance upon altering either
of these two fields. For evaluating the performance of the GPT-4 model, we
used the human annotated dataset as the ground truth for calculating metrics.

The model tends to perform better when it’s role is clearly defined. We
evaluated the model’s performance with three different role definitions as
shown in Table IV with the base definition being ”You are an intelligent
assistant”. For the other two roles, we gave the model more context and some
example labeled sentences in order to give it a better understanding of it’s
task. The model’s recall went up from 48.1% to 74.1% when the role was more
specific and had some examples for context.

However, the model still made a lot of false positive predictions with it’s
precision as low as 12.4%. For example, the sentence ”I have a big goal of
affording an Apple Vision Pro at this Christmas and the money I spend on
vaping would make that goal legit go up in smoke I want to look back on this
post and know; nicotine was never your friend.” was marked as positive (’Yes’)
by GPT-4. Although the speaker talks about distancing themselves from vaping,
there is no explicit or implicit mention of quitting vaping right now.

Subtle changes in words and sentence phrasing affect the way the model
responds to a prompt. One instance of hallucination was noted when the model
started evaluating every sentence in the context that it was related to
vaping. For example, it marked the sentence ”Quit!” as Yes and reasoned that
the word quit indicated that the speaker wanted to quit vaping. Not only does
that go against the annotation guidelines, but it shows yet another instance
of Generative AI models inferring data that isn’t there.

TABLE IV: GPT-4 Model Role Definitions Role | Change
---|---
You are an intelligent assistant | -
You are an intelligent assistant + Annotation guidelines used by human evaluators | added context
You are an intelligent assistant + Annotation guidelines used by human evaluators + Sample sentences with labels | added context, examples

###  IV-C Comparison

Before drawing any conclusions, it is important to note the limitations of
this study:

  * •

Training Data:

    * –

Size: For this preliminary study, we used a relatively smaller dataset to
train the BERT models. It is no secret that the models could have benefited
with a larger training corpus.

    * –

Diversity: All the posts were taken from the same subreddit community which
may not be representative of the discourse on other subreddits.

With a detailed and well-constructed prompt and precise role, GPT-4
demonstrated impressive capability to assist in social media data annotation.
However, caution must be taken to evaluate the model’s responses at each stage
of the pipeline to ensure that it is not biased. Key findings from our
comparison analysis indicate that the GPT-4 model was more consistent in
following annotation guidelines than the human evaluators. It was able to
detect implied quit vaping intentions that the human evaluators sometimes
missed.

##  V Conclusion and Future Work

With this preliminary study, we were able to compare the performance of
OpenAI’s GPT-4 model against traditional pre-trained BERT-based language
models and human evaluators on a data annotation task to identify vaping
cessation interests among Reddit users. We found that GPT-4 showed promising
results, sometimes even outperforming the human evaluators. However, caution
must be taken at each stage of the pipeline to minimize biases.

In the future, we plan to further develop this study as follows:

  * •

Larger Dataset: Using the phrases and sentences that mention quit-vaping
intentions, a much larger dataset can be collected through techniques such as
semantic search.

  * •

Multi-label Classification: This preliminary study only examined users’ quit-
vaping intentions at the sentence level due to the nature of work-in-progress.
Given the complexity of the social media data, we plan to include more vaping-
related labels such as negative health outcomes, reducing vape use, etc.

  * •

Contextual Learning: In this study, each sentence was evaluated individually
without the added context of the post that it was taken from. Machine Learning
models tend to perform better with more context, so we will expand this work
to contextual learning for the next-phase development.

## Acknowledgements

This research was generously funded by a NIH R34 grant (Grant No: CL040 155600
F1000 202 USCSP 10012339 1) and another research grant by the University of
South Carolina (USC) (PI: Dr. Dezhi Wu, Grant No: 80002838).

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11institutetext: Dublin City University
11email<EMAIL_ADDRESS>
# Neural Networks for Keyword Spotting on IoT Devices

Rakesh Dhakshinamurthy

###### Abstract

We explore Neural Networks (NNs) for keyword spotting (KWS) on IoT devices
like smart speakers and wearables. Since we target to execute our NN on a
constrained memory and computation footprint, we propose a CNN design that.
(i) uses a limited number of multiplies. (ii) uses a limited number of model
parameters.

##  1 Introduction

Nowadays the speech-related NNs are becoming exponentially popular since it is
an essential feature in a broad-spectrum of IoT devices. In IoT devices like
wearables and smart speakers, true hands-free speech recognition is achieved
by continuously listening to spot specific keywords. After spotting the
specified keywords, the local or online voice understanding model is woken up,
and the user’s voice is streamed to the model. This keyword spotting (KWS)
Models run on all the devices that are equipped with digital voice assistants
like Alexa. Even in the advanced voice-based systems and smart speaker
prototypes, such KWS models are integrated. For example, Sensory wake word
engine was used in the recent study [8]. Here, a face recognition algorithm
was trained using Deep Neural Network and deployed on their modern Alexa smart
speaker prototype. This model, without disturbing the smart speaker routine
can detect and identify a human face, then start the Alexa voice service only
when an authorized face is present in the live video frames. Similarly,
another recent study [9] also used the Sensory wake word engine to spot the
Alexa wake word. Here, a DL and Open CV based object detection model was
deployed in their smart speaker. After waking up the device by calling out the
Alexa command, whenever the user calls out _Alexa, ask Friday what she sees_ ,
the smart speaker camera turns on, then executes the deployed model and calls
out the names of detected objects as a response to the user command. The study
[7] presents a microphone array and voice algorithm-based smart hearing aid.
We recommend the authors to add a KWS spotting mechanism to the software
deployed in their prototype to improve the operating time of their device.

Since the above tiny devices and others have limited memory footprint and low
computational power, the current KWS systems need tuning to comfortably
execute on constrained setups and without disturbing the routine of the
devices. To efficiently deploy large models on such devices, we recommend
advanced model optimization methods such as _RC-NN_ [6] and _Edge2Train_ [5].
In the recent wearable-based study named _Avoid touching your face_ [10], the
authors have optimized their hand-to-face motion-detecting CNNs using RCE-NN
to enable the resource-friendly execution of the use-case model on a wide
range of smartwatches. Similarly, we recommend optimizing the adaptive
strategy model [4] to make it comfortably execute within the limited resource
of mobile IoT devices and carry out the model intended task of improving the
wireless communication quality of the target devices. Also, the parts of this
optimization pipeline are applicable to unsupervised methods [11].

In this paper, we take motivation from [1] to design CNNs for KWS use case. We
propose a CNN based approach since CNNs have shown better performance than
DNNs and also has a smaller model size. In our proposed design, we first limit
the overall computation of our KWS model. i.e., we limit the model parameters
and multiplies. Here, we first implement a CNN architecture that does not
perform pooling but rather performs filter striding in the frequency domain.
Second, we limit the total parameters during the design of the KWS model.

##  2 Keyword Spotting on IoT Devices

Figure 1: a. Shows the components of a KWS system (left to right) (i) Feature
Extraction (ii) Deep Neural Network (iii) Posterior Handling. b. Shows the
architecture of a typical CNN with a convolutional and max-pooling layer.

A block diagram of the KWS system [2] used in this work is shown in Fig. 1.
The first feature extraction component _computes 40-dimensional log-mel
filterbank features every 25ms with a 10ms frame shift_. Next, in each frame,
we stack 25 frames to the left and 10 frames to the right and provide this as
an input to DNN which is the second component. This DNN architecture has 3
hidden layers with 128 hidden units plus a softmax layer, where each hidden
layer uses a ReLU nonlinearity. The softmax is the output layer containing one
target output for each of the words that need to be detected from the audio
signal stream. This model’s weights are trained to optimize a cross-entropy
criterion using the distributed asynchronous gradient descent. The final
component is the posterior handling module. Here for each frame, posterior
scores from the DNN are combined into a single score.

##  3 Keyword Spotting CNN Design

Here we present a CNN design that is our proposed alternative to the DNN
described in the previous Section.

###  3.1 Typical CNN Architecture

Figure 2: Traditional CNN Architecture used for the KWS task. Par. is the
number of parameters and Mul. is the number of multipliers used by each part
of the CNN.

A typical convolutional architecture shown in Fig. 1 has been heavily used on
many challenges [3]. When the log-mel input ($t\texttimes f=32\texttimes 40$)
is fed into the model, the first layer has a filter size with a specified
frequency (freq $r=9$). The common method is to choose a filter size in time
spanning two-thirds of the overall input size. Multiplication is performed by
striding this filter by $s=1$ and $v=1$ across both frequency and time
domains. Next, non-overlapping max-pooling is performed with a pooling region
of $q=3$. The second filter has a filter of $r=4$ in frequency, and no max-
pooling is performed. Since we target limited resource devices, we need to
keep the number of parameters as low as possible. We show the architecture of
a traditional CNN in Fig. 2. Here, as shown, the model has 2 convolutional,
one linear low-rank, and one DNN layer. Currently, this architecture has a
huge number of multiplies in the convolutional layers due to the 3-D inputs
spanning across all the three frequency, time, and feature maps. Hence, not
feasible for power-constrained small-footprint devices where multiplies are
limited. In the below sections, we present alternative CNN architectures to
address the tasks of limiting parameters or multiplies.

###  3.2 Limiting Multiplies

Figure 3: Optimized CNN Architecture that performs the same KWS task but has a
lesser number of Parameters and Multipliers than original CNN.

To obtain a resource-friendly CNN we need to limit the number of multiplies.
One solution to limit multiplies is to have a single convolutional layer, and
also have the time filter span all of the time. The output of this
convolutional layer should be passed to a linear low-rank layer followed by
two DNN layers. Fig. 3 shows an architecture with only one convolutional
layer. Here, the number of multiplies is cut by a factor of 10, compared to
the CNN from Fig. 2.

###  3.3 Limiting Model Parameters

Increasing model parameters leads to performance improvements at the cost of
model size. In this section, we explore CNN architectures by limiting the
model size to 250K but we do not limit the multiplies like in the previous
section. We can improve performance by increasing the input features. Keeping
parameters fixed by doing so requires exploring sampling of input audio signal
in time and frequency. When designing models for acoustic use cases, the
called words that we want to classify occur over within a very short time-
duration, making pooling reduce the model detection accuracy. But the keywords
like Alexa occur over a longer time-span (50-100ms). Thus, we propose to
improve the resource friendliness of models by sub-sampling the signal in time
using striding or pooling. In the future, we shall explore the conventional
sub-sampling in time with longer acoustic units.

##  4 Conclusion

In this paper, we presented a CNN design that is best suited for the KWS use
case. When comparing CNNs with DNNs, when we limit the number of multiplies
and model parameters of CNNs, we hypothesize to achieve performance
improvements in both clean and noisy environments. In the future, we shall
evaluate our proposed design and open-source the trained model.

## References

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# Bulk Majorana mass terms and Dirac neutrinos in Randall Sundrum Model

Abhishek M Iyer<EMAIL_ADDRESS>Centre for High Energy Physics,
Indian Institute of Science, Bangalore 560012  Sudhir K Vempati
<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of
Science, Bangalore 560012

###### Abstract

We present a novel scheme where Dirac neutrinos are realized even if lepton
number violating Majorana mass terms are present. The setup is the Randall-
Sundrum framework with bulk right handed neutrinos. Bulk mass terms of both
Majorana and Dirac type are considered. It is shown that massless zero mode
solutions exist when the bulk Dirac mass term is set to zero. In this limit,
it is found that the effective 4D small neutrino mass is primarily of Dirac
nature with the Majorana type contributions being negligible. Interestingly,
this scenario is very similar to the one known with flat extra dimensions.
Neutrino phenomenology is discussed by fitting both charged lepton masses and
neutrino masses simultaneously. A single Higgs localised on the IR brane is
highly constrained as unnaturally large Yukawa couplings are required to fit
charged lepton masses. A simple extension with two Higgs doublets is presented
which facilitates a proper fit for the lepton masses.

###### pacs:

73.21.Hb, 73.21.La, 73.50.Bk

1. As of today, we have no experimental indication whether neutrinos are of Majorana type or the Dirac type. On the theoretical side, models of both Dirac and Majorana type have been considered. The popular seesaw mechanism with lepton number violation in the right handed neutrino sector Mohapatra  leads to small Majorana type masses for the left handed neutrinos. Dirac type neutrinos on the other hand traditionally require lepton number conservation. In most models either this conservation is imposed by hand/construction or is a residue of some larger flavour symmetry 10 ; 11 ; 12 ; 13 ; gross ; 15 ; Dienes ; 17 ; 19 ; 20 ; 21 ; 22 ; 23 ; Memenga . Conservation of global quantum numbers like lepton number is typically disfavored theoretically due to arguments based on quantum gravity and worm holes Witten . For this reason, Dirac neutrinos are considered to be some what unnatural.

One possibility could be that lepton number is violated only by Planck scale
operators. If these operators are then some how suppressed, this would
naturally pave way for Dirac neutrino masses 111It should be noted that an
alternate approach would be to consider discrete flavour symmetries which are
imposed to avoid Majorana mass terms and thus leading to Dirac neutrino masses
Aranda:2013gga .. In four dimensions the impact of the lepton number violation
at the Planck scale is characterized by the effective operator $LH.LH/M_{Pl}$
at the weak scale. This leads to corrections to the neutrino mass matrix
$\sim~{}\mathcal{O}(10^{-3})~{}\text{eV}$ if one assumes $\mathcal{O}(1)$
coefficients. In higher dimensions explicit constructions can be done with
specific Planck scale lepton number violating operators and their impact on
weak scale physics can be studied. In fact in a particular example presented
in planckgher  it has been shown that lepton number violation at the Planck
scale can almost be hidden from weak scale physics.

In this case, a Randall-Sundrum (RS) RS  setup with two branes located at the
two orbifold fixed points is considered. The two fixed points, located at the
$y=0$ and $y=\pi R$ are identified with the $UV\sim M_{Pl}$ and the
$IR\sim\text{TeV}$ scales respectively. The line-element for the RS background
is given as

$ds^{2}=G_{MN}dx^{M}dx^{N}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$
(1)

where $\sigma(y)=k|y|$. Fermions and gauge bosons are allowed to propagate in
the bulk while the Higgs is localized on the IR brane It has been shown that
in the limit when the right handed neutrinos are highly IR localized, whereas
the Majorana mass terms are localised on the Planck brane, lepton number
violating effects in effective neutrino mass matrix in four dimensions are
highly suppressed. Neutrino masses can be of Dirac type by localised operators
on the IR brane planckgher . The main idea here being that the geometric
‘separation’ of the fields and the lepton number violating operators leads to
suppressed effects of the latter in the effective neutrino mass matrix. In the
present letter we present a novel way of realizing Dirac masses in the same RS
setup. We will consider dimensionful Majorana mass terms as lepton number
violating operators. In particular we show that the lepton number violating
operators need not be localized on the UV brane but instead can be present in
the bulk. The magnitude of the operators can be as large as the Planck scale.
The 4D neutrinos are almost Dirac like, in the limit when the bulk Dirac mass
terms for the right handed neutrinos are set to zero. Interestingly this case
is very similar to the case discussed in flat extradimensions.

2. Extra space dimensions offer a new way of looking at the fermion mass hierarchy in the SM. Fermion bulk wave-functions are ‘split’ and are localized at different points in the extra-dimension ArkaniHamed . The point of localization is determined by the bulk Dirac mass parameters introduced separately for the left and right components of the 5D matter fields. The overlap of the zero mode wave-functions with the Higgs field determines the effective four dimensional Yukawa coupling of the fermion. In the Randall-Sundrum setup, where the bulk geometry is warped, localization of the fermions is natural; the point of localization is again determined by the bulk Dirac mass terms. Neutrinos are however different from other matter fields as they allow both Dirac and Majorana mass terms. The profile of the zero mode crucially depends on the interplay between these two terms and the boundary conditions one chooses.

We now give a brief review of bulk (right handed) neutrino fields in the RS
set up. This case has been discussed in several papers warpedseesaw ; a ; b ;
c ; d ; planckgher ; perez ; Watanabe ; iyer . The leptonic and the quark part
of the action is given by

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle S_{Kinetic}+\int
d^{4}x\int
dy\sqrt{-g}\left[\frac{1}{2}\left(m_{M}\bar{N}N^{c}+\text{h.c.}\right)+\ldots\right.$
$\displaystyle+$
$\displaystyle\left.\left(Y_{N}\bar{L}\tilde{H}N+Y_{E}\bar{L}EH+\ldots\right)~{}\delta(y-\pi
R)\right]$ $\displaystyle S_{Kinetic}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\int
dy~{}\sqrt{-g}~{}\left(~{}\bar{Q}(i\not{D}-m_{Q})Q+\bar{u}(i\not{D}-m_{u})u+\bar{d}(i\not{D}-m_{d})d\right.$
(2) $\displaystyle+$
$\displaystyle\left.\bar{N}(i\not{D}-m_{N})N+\bar{L}(i\not{D}-m_{L})L+\bar{E}(i\not{D}-m_{E})E~{}\right).$

where the covariant derivative is defined ass

$D_{M}=\partial_{M}+\Omega_{M}+\frac{ig_{5}}{2}\tau^{a}W_{M}^{a}(x,y)+\frac{ig^{\prime}}{2}Q_{Y}B_{M}(x,y)$
(3)

with $\Omega_{M}=(-k/2e^{-ky}\gamma_{\mu}\gamma^{5},0)$ being the spin
connection and $Q_{Y}$ is the hypercharge. $M$ is the five dimensional Lorentz
index. In the above, $N$ ( $E$ ) are the neutrino (charged lepton ) singlet
fields, $L$ are the lepton doublet fields and the Higgs field, is denoted by
$H$ with $\tilde{H}=i\sigma_{2}H^{*}$. Generation indices have been
suppressed. The bulk mass parameters for the $N$ fields are $m_{M}$ ($m_{N}$)
for the Majorana (Dirac) type. Here $N^{c}=C_{5}\bar{N}^{T}$ with $C_{5}$
being the five-dimensional charge conjugation matrix222$C_{5}$ is taken to be
$C_{4}$.. $m_{L}$($m_{E}$) stands for the bulk (Dirac type) mass terms of the
doublet (singlet) fields. All the bulk mass parameters are expressed in terms
of the so called $c$ parameters, for example, $m_{M}=c_{M}k$, with $k$ being
the reduced Planck scale. Similarly, $m_{N}$ = $c_{N}k$ etc333In the
following, we will consider all the mass parameters to be real.. $Y_{N,E}$ are
the Yukawa parameters with mass dimensions, $[Y]=-1$. Finally, let us note
that in the above action we assumed the Higgs field to be localized on the IR
brane.

The bulk fields can be Kaluza-Klein (KK) expanded in terms of their four
dimensional fields and profiles in the fifth direction. For the discussion
relevant here, we consider the expansions of $N$ and $L$ fields as:

$\displaystyle N(x,y)=\sum_{n=0}^{\infty}{e^{2\sigma(y)}\over\sqrt{\pi
R}}\left(N^{(n)}_{1}(x)g_{1}^{(n)}(y)+N^{(n)}_{2}(x)g_{2}^{(n)}(y)\right)$
$\displaystyle L(x,y)=\sum_{n=0}^{\infty}{e^{2\sigma(y)}\over\sqrt{\pi
R}}\left(L^{(n)}_{L}(x)f_{L}^{(n)}(y)+L^{(n)}_{R}(x)f_{R}^{(n)}(y)\right)$ (4)

where $N_{1}^{(n)}(x)$ and $N_{2}^{(n)}(x)$ are the two Weyl components of the
neutrino singlet field with $g_{1}^{(n)}(y)$ and $g_{2}^{(n)}(y)$ representing
their profiles in the $y$ direction444We will specify the $Z_{2}$ properties
of these components separately for each case we consider.. Similarly, for the
$L$ field, $L_{L}^{(n)}(x)$ and $L_{R}^{(n)}(x)$ represent the Weyl components
along with $f_{L}^{(n)}(y)$ and $f_{R}^{(n)}(y)$ represent the respective
profiles. The profiles can be derived from the action after imposing the
ortho-normality conditions. For the KK modes of the $N$ field, the profiles
are the solutions of the following couple differential equations warpedseesaw

$\displaystyle(\partial_{y}+m_{N})g_{1}^{(n)}(y)=m_{n}e^{\sigma}g_{2}^{(n)}(y)-m_{M}g_{2}^{(n)}(y)$
$\displaystyle(-\partial_{y}+m_{N})g_{2}^{(n)}(y)=m_{n}e^{\sigma}g_{1}^{(n)}(y)-m_{M}g_{1}^{(n)}(y)$
(5)

A crucial point to note is that zero mode solutions, $m_{n}=0$ in the set of
equations in Eq.(5), are not consistent with the boundary conditions at the
orbifold fixed pointswarpedseesaw . Solutions however do exist for higher
modes and they can be obtained numerically. A detailed phenomenological
analysis can be found in iyer .

3. Let us now revisit the result of Ref.planckgher  where Dirac neutrinos are realized in the above RS setup with Majorana operators. Zero mode solutions for Eq.(5) are however possible if the Majorana mass terms are localized on the UV or IR boundary. For the case where they are confined to the UV boundary, the bulk eigenvalue equations for the $N$ fields in Eq.(5) simply reduce to

$\displaystyle(\partial_{y}+m_{N})g_{1}^{(n)}(y)=m_{n}e^{\sigma}g_{2}^{(n)}(y)$
$\displaystyle(-\partial_{y}+m_{N})g_{2}^{(n)}(y)=m_{n}e^{\sigma}g_{1}^{(n)}(y)$
(6)

Analytical solutions can easily be derived for $m_{n}=0$. Lets consider
$N_{1}$ component to be even under the $Z_{2}$ symmetry and $N_{2}$ component
to be odd. The profile of $N_{1}^{(0)}=N_{1}$ is given by
$g_{1}^{(0)}(y)=g_{1}(y)=\mathcal{N}_{N}e^{-c_{N}ky}$, where as the $Z_{2}$
odd field, $N_{2}$, has no zero modes. The normalization factor
$\mathcal{N}_{N}$ is given as

$\mathcal{N}_{N}=\sqrt{\frac{0.5-c_{N}}{\epsilon^{2c_{N}-1}-1}}$ (7)

where $\epsilon=e^{-kR\pi}$. For a regular RS setup $kR\sim 11.4$, which
implies $\epsilon\sim 10^{-16}$. It should be noted that profiles of the zero
modes of $L,E$ fields also carry the same form. The zero mode of $N_{1}$ field
picks up a Majorana mass due to the localised term at the UV boundary given
by:

$\displaystyle m_{N^{(0)}}$ $\displaystyle\sim$ $\displaystyle
m_{M}g_{1}(0)^{2}$ (8)

Assuming $c_{N}<0.5$ and $m_{M}\sim k$, the above equation becomes

$\displaystyle m_{N^{(0)}}$ $\displaystyle\sim$ $\displaystyle
k~{}(0.5-c_{N})e^{-(1-2c_{N})kR\pi}$ (9) $\displaystyle\sim$ $\displaystyle
1~{}\text{TeV}~{}\epsilon^{-2c_{N}}$

where $k\epsilon\sim 1~{}\text{TeV}$. To analyze the neutrino mass matrix, we
should also consider the IR brane localised terms, the second line of Eq.(Bulk
Majorana mass terms and Dirac neutrinos in Randall Sundrum Model), generated
from the Yukawa interaction. These are Dirac mass terms and are given by

$m_{D_{\nu}}^{(0,0)}=\frac{v}{\sqrt{2}}g_{1}(\pi R)Y_{N}^{\prime}f_{L}(\pi R)$
(10)

where the $\mathcal{O}$(1) parameter $Y_{N}^{\prime}=2kY_{N}$ and
$f_{L}^{(0)}=\mathcal{N}_{L}e^{-c_{L}ky}$ denotes the zero mode profile of the
doublet. The resultant neutrino mass matrix (with one KK mode ) has Type-I
seesaw structure. In the basis $\eta^{T}=\\{{\nu_{L}^{(0)},N_{1}^{(0)}}\\}$,
the Majorana mass to the lowest order is given as

$\mathcal{L}_{m}=-{1\over
2}\eta^{T}\mathcal{M}_{\nu}\eta\;\;\;;\;\;\;\;\mathcal{M}_{\nu}=\begin{pmatrix}0&m_{D_{\nu}}\\\
m_{D_{\nu}}&m_{N^{(0)}}\end{pmatrix}$ (11)

where we have assumed one flavour for simplicity. From Eq.(9) we see that as
$c_{N}\rightarrow-\infty$, $m_{N^{(0)}}\rightarrow 0$ 555Note that
$c_{N}\sim-1$ is sufficient to make $m_{N^{(0)}}$ insignificant.. Note that
this limit holds while $c_{M}$ is taken to be $\mathcal{O}$(1) and the
Majorana mass terms can be $\mathcal{O}(M_{PL})$. As a result the Majorana
mass for the right handed neutrino almost vanishes. In this limit, the
eigenvalues of the neutrino mass matrix in Eq.(11) are $\pm
m_{D_{\nu}}^{(0,0)}$. This implies that the localization of the zero mode of
the neutrino singlet very close to the IR brane results in its negligible
overlap with the lepton number violating operator situated on the UV brane.
The small neutrino masses are determined entirely by the brane localized
Yukawa coupling in Eq.(Bulk Majorana mass terms and Dirac neutrinos in Randall
Sundrum Model) thus attributing a Dirac nature to the neutrinos.

4. We now discuss the alternative possibility of realizing Dirac neutrinos in the presence of lepton number violating terms. However we will assume $m_{N}=0$ in Eq.(Bulk Majorana mass terms and Dirac neutrinos in Randall Sundrum Model). Bulk flavour symmetry groups can be imposed to achieve this limit. For example consider the following bulk flavour symmetry group for leptons

$G_{lepton}\equiv SU(3)_{L}\times SU(3)_{E}\times O(3)_{N_{R}}$ (12)

The transformation of the leptonic fields under $G_{lepton}$ is given as

$\displaystyle L\sim(3,1,1)\;\;\;E\sim(1,3,1)\;\;\;\
N_{R}\sim(1,1,3)\;\;\;\;N_{L_{i}}\sim(1,1,1)~{}(i=1,2,3)$ (13)

Note that we have given different representations for the left and right
chiralities of the $N$ field. The $Z_{2}$ odd field $N_{L}$ is considered to
transform as a singlet under $O(3)$. This choice leads to a vanishing bulk
Dirac mass ($m_{N}=0$) for the singlet field. In this case, the Majorana mass
terms are no longer localised on the UV brane, but are present in the bulk. In
this limit where the bulk Dirac mass for $N$ vanishes, the solutions to Eq.(5)
are simple to obtain and are given as

$\displaystyle g_{1}^{(n)}(y)$ $\displaystyle=$
$\displaystyle\xi\sin(\frac{m_{n}e^{\sigma}}{k}-m_{M}y),$ $\displaystyle
g_{2}^{(n)}(y)$ $\displaystyle=$
$\displaystyle\xi\cos(\frac{m_{n}e^{\sigma}}{k}-m_{M}y)$ (14)

where $\xi\sim\sqrt{\pi Rk}e^{-0.5\sigma(\pi R)}$. Imposing proper boundary
conditions we can project out the $Z_{2}$ odd components on the boundary. For
example, we choose as before that the $N_{2}$ component is $Z_{2}$ odd and
$N_{1}$ is the $Z_{2}$ even component. The boundary conditions for which the
$Z_{2}$ odd part say $g_{2}$ vanishes on the boundary are given as

$\frac{m_{n}e^{\sigma}}{k}-m_{M}\pi R=(2n+1){\pi\over 2}$ (15)

where n=0,1,2…. The zero mode666In principle massless modes are also possible
by choosing $m_{M}=\frac{-p}{R}$ where $p\in\mathcal{Z}^{+}$.($n=0$, massless
solutions) can exist if $m_{M}$ takes values $m_{M}=\frac{-1}{2R}$. In this
case, we have

$\displaystyle g_{1}^{(0)}(y)$ $\displaystyle=$
$\displaystyle\xi\sin(-m_{M}y)$ $\displaystyle g_{2}^{(0)}(y)$
$\displaystyle=$ $\displaystyle\xi\cos(m_{M}y)$ (16)

This corresponds to $c_{M}=\frac{-1}{2kR}$. The masses of the higher KK modes
are determined from the boundary conditions in Eq.(15) and are given as

$m_{n}\sim nk\pi\epsilon\;\;\;;\;n=1,2,3\ldots$ (17)

Unlike the other RS fields, the KK modes for the $N$ field are regularly
spaced at $1\pi$ TeV, $2\pi$ TeV etc. This reminds us of the KK bulk fields in
Arkani-Hamed, Dimopouplos, Dvali (ADD) models ADD ; Antoniadis:1998ig . Thus
if one considers bulk Majorana mass terms instead of Dirac mass terms, the
profiles in the bulk for the zero and higher modes are periodic.

Let us consider the total neutrino mass matrix in this case. In the basis,
$\chi^{T}=\\{\nu_{L}^{(0)},N^{(0)}_{1},N^{(1)}_{1}\ldots\\}$ the neutrino mass
matrix takes the form

$\mathcal{L}_{m}=-{1\over
2}\chi^{T}\mathcal{M}\chi\;\;\;;\;\;\;\;\mathcal{M}=\begin{pmatrix}0&m_{D_{\nu}}^{(0,0)}&-m_{D_{\nu}}^{(0,1)}&m_{D_{\nu}}^{(0,1)}&\ldots\\\
m_{D_{\nu}}^{(0,0)}&0&0&0&\ldots\\\ -m_{D_{\nu}}^{(0,1)}&0&m_{1}&0&\ldots\\\
m_{D_{\nu}}^{(0,1)}&0&0&m_{2}&\ldots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{pmatrix}$ (18)

where the 4D Dirac mass for the neutrino induced on the IR brane is given as

$m_{D_{\nu}}^{(0,0)}=\frac{v}{\sqrt{2}}Y^{\prime}_{N}\sqrt{\frac{0.5-c_{L}}{\epsilon^{2c_{L}-1}-1}}\epsilon^{c_{L}-0.5}$
(19)

where $Y^{\prime}_{N}=2kY_{N}$ and
$|m_{D_{\nu}}^{(0,0)}|=|m_{D_{\nu}}^{(0,i)}|$ $\forall$ $i\geq 1$. The form of
this mass matrix is very similar to the one with bulk right singlet neutrinos
and brane localised left handed doublet lepton fields in flat extra dimensions
considered in Ref.Dienes . The analysis of Ref. Dienes  can be used to find
the eigenvalues of the mass matrix (18).

If there are $n_{0}$ KK modes in the theory, the effective neutrino mass
matrix in the basis $(\nu_{L}^{(0)},N^{(0)}_{1})$, after integrating our the
KK modes is given as Dienes

$\mathcal{M}_{eff}=\begin{pmatrix}a_{0}&m_{D_{\nu}}^{(0,0)}\\\
m_{D_{\nu}}^{(0,0)}&0\end{pmatrix}$ (20)

where $a_{0}\sim-\frac{m^{2}R}{\epsilon}\ln{n_{0}}$. The neutrino mass
eigenvalues can be written as

$m_{\nu}\sim\pm m_{D_{\nu}}^{(0,0)}-\frac{m^{2}R}{\epsilon}\ln{n_{0}}$ (21)

In the limit where $a_{0}~{}\ll~{}m_{D_{\nu}^{(0,0)}}$, neutrinos are
automatically Dirac-like. The second term in Eq.(21) is the Majorana “seesaw”
like terms which is a result of integrating out heavy KK modes. In contrast
with the ADD case, this limit is natural in the RS case. In the RS case, $R$
is small, $\sim{1\over k}$, which makes the contribution from the Majorana
term negligible. This result holds even for a large $n_{0}\sim 10^{18}$. On
the other hand, in the ADD case, a very large radius, $R\sim\text{eV}^{-1}$ is
required to have a large contribution from the “seesaw” term in the limit
$n_{0}$ becomes very large. $m_{D_{\nu}}^{(0,0)}$ can be made small $\sim
m_{atm}$ by choosing $c_{L}$ values appropriately. For example, this can be
achieved by localizing the leptonic doublets close to the UV brane.

Finally let us note that the above situation can be easily generalized to
three generations with three bulk right handed neutrinos. In the next section
we discuss neutrino phenomenology in detail.

5. We simultaneously fit neutrino mass differences and charged lepton masses along with the mixing angles to the $c$ parameters and the $\mathcal{O}(1)$ Yukawa couplings. More details of the fit can be found in iyer . For the standard RS set up, $\epsilon\sim 10^{-16}$, we find that to reproduce the atmosphere neutrino mass scale, $\mathcal{O}$(0.03) eV we need a $c_{L}\sim 1.3$. We have chosen the $\mathcal{O}(1)$ Yukawa coupling $Y^{\prime}$ to be 1. Such $c$ values for the lepton doublets make it difficult to fit simultaneously the charged lepton masses777Such a situation was also encountered in the case neutrinos get their masses through higher dimensional lepton number violating operator iyer .. The charged lepton mass matrix is given as

$m_{ij}=(Y^{\prime}_{E})_{ij}\mathcal{N}_{L_{i}}\mathcal{N}_{E_{j}}\epsilon^{c_{L_{i}}+c_{E_{j}}-1}$
(22)

where $\mathcal{N}_{L,E}$ have the same form as Eq.(7) and the dimensionless
$\mathcal{O}(1)$ Yukawa coupling for a brane localized Higgs is defined as
$Y^{\prime}_{E}=2kY_{E}$. The required bulk mass parameters for the charged
lepton fields, $c_{E}$ turn out to be large and negative. This introduces a
host of other problems like non-perturbative Yukawa couplings etc. On the
other hand, changing the warp factor will not have much impact on the results.
For example, for a $\epsilon\sim 10^{-2}$, we find that $c_{L}$ values
required are even larger.

A simple solution would be to disentangle the Higgs fields responsible for
neutrino masses and charged leptons by introducing an additional Higgs doublet
as in a two Higgs doublet model. We denote the Higgs doublet generating the
Dirac neutrino masses by $H_{u}$ and the other as $H_{d}$. $H_{u}$ is
localised on the IR brane where as the $H_{d}$ is a bulk Higgs field, whose
localisation is fixed by the charged lepton masses. The Yukawa part of the
lagrangian is now given as

$L_{Yuk}\subset\int d^{4}xdy\left[\left(\delta(y-\pi
R)Y_{N}\bar{L}H_{u}N+Y_{E}\bar{L}H_{d}E\ldots\right)\right]$ (23)

As before, the neutrino masses are given by Eq.(19) with $H$ replaced by
$H_{u}$. To determine the charged lepton mass matrices with a bulk Higgs, we
briefly review the the derivation for the profile equation as well as the
Yukawa couplings for a bulk Higgs.

For a bulk scalar field in a warped background, the presence of zero modes
requires the addition of brane localized mass terms. For the bulk field
$H_{d}$, the action is given as gherghetta ; Gherghetta1

$S=\int
d^{4}xdy\sqrt{-g}\left[\partial_{M}H_{d}^{*}\partial^{M}H_{d}+\left[m_{H_{d}}^{2}+2bk\left(\delta(y)-\delta(y-\pi
R)\right)\right]|H_{d}|^{2}\right]$ (24)

where we parametrize the bulk mass as $m_{H_{d}}^{2}=ak^{2}$ with $a,b$ being
dimensionless quantities. Ideally one would expect them to be
$\mathcal{O}$(1). The zero mode profile for a bulk scalar is given as

$f_{H_{d}^{(0)}}=\sqrt{k\pi R}\zeta_{H_{d}}e^{(b-1)ky}$ (25)

where the normalization factor $\zeta_{\phi}$ is given as

$\zeta_{H_{d}}=\sqrt{\frac{2(b-1)}{\epsilon^{2(1-b)}-1}}$ (26)

The brane parameter $b$ must be tuned to be $b=2\pm\sqrt{4+a}$ to satisfy the
boundary conditions for the zero modes. $b>1(b<1)$ implies the zero mode of
the Higgs is localized towards the IR(UV) brane.

For a bulk Higgs the fundamental Yukawa couplings $Y_{E}$ have mass dimension
-1/2. After performing the KK expansion and integrating over the extra-
dimension the zero mode mass matrix for all charged leptons in general is
given as 888The down type quarks have the same form of the mass matrix as the
charged leptons while the up type quark mass matrix is similar to Eq.(22).

$m_{ij}=v_{d}(Y^{\prime}_{E})_{ij}\zeta_{H_{d}}\mathcal{N}_{{L_{i}}}\mathcal{N}_{{E_{j}}}\left(\frac{\epsilon^{(c_{L_{i}}+c_{E_{j}}-b)}-1}{b-c_{L_{i}}-c_{E_{j}}}\right)$
(27)

where we have defined the dimensionless $\mathcal{O}$(1) Yukawa coupling as
$Y^{\prime}_{E}=2\sqrt{k}Y_{E}$ and the normalization factor $\mathcal{N}_{i}$
is defined in Eq.(7). The corresponding $\mathcal{O}$(1) parameters for the up
and the down sector quarks are denoted as $Y^{\prime}_{U}$ and
$Y^{\prime}_{d}$ respectively.

The ratio of the vev of the two Higgs doublet is defined as
$tan\beta=\frac{v_{u}}{v_{d}}$. We choose $tan\beta=10$ for illustration.
While $H_{u}$ is localized on the IR brane, $H_{d}$ is localized near the UV
brane ($b<1$). Fitting Eq.(19) for small neutrino masses require $c$ value for
the doublet to be $\sim 1.3$. Corresponding to this and fitting Eq.(27) for
the charged leptons we choose $b=0.3$. We find that the electron mass can be
conveniently fit by choosing $c_{E_{R}}\sim 0.3$ while the remaining charged
leptons can be fit by choosing a range $0.4<c<1$ for the corresponding bulk
mass parameters of the singlets. Table[1] shows the range of $c$ parameters
obtained which fit the lepton masses and mixing angles. For the leptonic case
we assume normal hierarchy of neutrino mass eigenvalues. The magnitude of
$\mathcal{O}(1)$ Yukawa parameters in the leptonic sector i.e.
$Y^{\prime}_{N,E}$ were chosen to be in the range $[0.08,4]$ to fit the data.
This configuration of Higges can also accommodate quark masses by the
introduction of 9 bulk mass parameters i.e $c_{Q},c_{U},c_{d}$ in Eq.(Bulk
Majorana mass terms and Dirac neutrinos in Randall Sundrum Model). To fit the
top quark mass, the third generation top singlet require $c_{t_{R}}\leq-3.0$
while the lighter generations including the third generation doublet can be
fit by choosing the corresponding $c$ values to be $0.3<c<1$. Table[2] shows
the range of $c$ parameters for the hadronic sector which fit the quark masses
and CKM angles. The magnitude of $\mathcal{O}$(1) Yukawa parameters for the
quark sector were also chosen to lie between $0.08<|Y^{\prime}_{a}|<4$ where
$a=d,U$.

parameter | range | parameter | range
---|---|---|---
$c_{L_{1}}$ | 1.27-2.4 | $c_{E_{1}}$ | 0.19-0.36
$c_{L_{2}}$ | 1.26-1.4 | $c_{E_{2}}$ | 0.35-0.40
$c_{L_{3}}$ | 1.25-1.38 | $c_{E_{3}}$ | 0.44-0.52
Table 1: Range of $c$ parameters which fit the lepton masses and mixing angles in the bulk two Higgs doublet model with bulk Majorana masses. $\mathcal{O}$(1) Yukawa parameters are chosen to lie between $0.08<|Y^{\prime}|<4$. Normal hierarchy is assumed for the neutrino masses. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{Q_{1}}$ | 0.39-0.57 | $c_{D_{1}}$ | 0.30-0.49 | $c_{U_{1}}$ | 0.67-0.79
$c_{Q_{2}}$ | 0.47-0.55 | $c_{D_{2}}$ | 0.37-0.93 | $c_{U_{2}}$ | 0.53-0.58
$c_{Q_{3}}$ | 0.502-0.507 | $c_{D_{3}}$ | 0.68-0.97 | $c_{U_{3}}$ |  $-6.8$ \- $-3.0$
Table 2: Range of $c$ parameters which fit the quark masses and mixing angles
in the bulk two Higgs doublet model with bulk Majorana masses.
$\mathcal{O}$(1) Yukawa parameters are chosen to lie between
$0.08<|Y^{\prime}|<4$.

6. Nature has not yet spoken whether neutrinos are Dirac or Majorana. While theoretically we are prejudiced to consider that Majorana neutrinos are more natural as quantum gravity does not conserve global symmetries, it is not uncommon to find examples where Dirac neutrinos can exist even with lepton number violation. In the present work, using the RS set up, we presented a scenario in which Dirac neutrinos can be obtained in the presence of lepton number violating terms in the bulk.

We have considered lepton localisation purely with bulk Majorana mass terms.
Bulk Dirac mass terms are set to zero. This case leads to periodic KK modes
similar to ADD models. Zero modes can exist for particular values of the
Majorana mass terms. In this case, the Majorana contribution can be shown to
be negligible leading to Dirac neutrinos. Phenomenologically, this model
requires large $c_{E}$ values for the bulk mass parameters which is
problematic to fit charged lepton masses. A simple extension in terms of two
Higgs doublet model is presented where a good fit to neutrino masses and
charged lepton masses is simultaneously obtained. We have not commented about
electroweak precision constraints nor flavour constraints in this model as our
focus has been purely on fermion masses. Electroweak precision constraints are
expected to be strong and one possible way out is to consider a much larger
gauge group and particle spectrum towards a custodially symmetric RS model
Agashe:2003zs . We expect that inclusion of both the Higgses should be
straightforward. On the other hand flavour is expected to be strong due to
presence of new Higgs contributions. Fortunately, one can utilise Minimal
Flavour Violation (MFV) techniques to reduce flavour violation. For the
flavour symmetry group presented in Eq.(12), the Yukawa couplings transform as

$Y_{E}\sim(3,1,\bar{3})\;\;\;\;\;Y_{N}\sim(3,1,3)$ (28)

The bulk mass parameters can be expressed in terms of the Yukawa as
Fitzpatrick

$c_{L}=I+\alpha
Y_{E}Y_{E}^{\dagger}+\alpha^{\prime}Y_{N}Y_{N}^{\dagger}\;\;\;\;\;c_{E}=\beta
Y_{E}^{\dagger}Y_{E}\;\;\;\;\;c_{N}=0$ (29)

where $\alpha,\alpha^{\prime},\beta\in\mathcal{R}$. This leads to a strong
suppression of flavour violation. A detailed analysis of flavour violation in
RS models with two Higgs doublets will be presented elsewhere ourrs2 . The
investigation of the fermion mass hierarchy in two Higgs doublets models and
the resulting implications is interesting enough to be considered in greater
detail.

Acknowledgement

SKV acknowledges support from DST Ramanujam fellowship SR/S2/RJN-25/2008 of
Govt. of India.

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