# Factual and Informative Review Generation
for Explainable Recommendation

Zhouhang Xie Julian McAuley Bodhisattwa Prasad Majumder
University of California, San Diego
{zhx022, bmajumde<EMAIL_ADDRESS>Zhouhang Xie,1 Sameer Singh,2 Julian
McAuley, 1 Bodhisattwa Prasad Majumder 1

###### Abstract

Recent models can generate fluent and grammatical synthetic reviews while
accurately predicting user ratings. The generated reviews, expressing users’
estimated opinions towards related products, are often viewed as natural
language ‘rationales’ for the jointly predicted rating. However, previous
studies found that existing models often generate repetitive, universally
applicable, and generic explanations, resulting in uninformative rationales.
Further, our analysis shows that previous models’ generated content often
contain factual hallucinations. These issues call for novel solutions that
could generate both informative and _factually grounded_ explanations.
Inspired by recent success in using retrieved content in addition to
parametric knowledge for generation, we propose to augment the generator with
a personalized retriever, where the retriever’s output serves as external
knowledge for enhancing the generator. Experiments on Yelp, TripAdvisor, and
Amazon Movie Reviews dataset show our model could generate explanations that
more reliably entail existing reviews, are more diverse, and are rated more
informative by human evaluators.

##  1 Introduction

Recently, there has been increasing interest in treating review generation as
a proxy for explainable recommendation, where generated reviews serve as
rationales for the models’ recommendations(Li et al. 2016; Li, Zhang, and Chen
2021; Ni et al. 2017; Ni and McAuley 2018). However, existing models commonly
generate repetitive and generic content, resulting in uninformative
explanations (Geng et al. 2022). Further, when evaluating the factuality of
generated reviews using pre-trained entailment model, our analysis shows that
existing models are also susceptible to factual hallucination, a long-existing
challenge in many natural language generation (NLG) tasks (Pagnoni,
Balachandran, and Tsvetkov 2021; Maynez et al. 2020). Specifically, the models
often generate statements that are not supported by information about the
corresponding product in the training set. Both nonfactual and uninformative
explanations are undesirable, as end users would look for recommendation
rationales that truthfully reflect the characteristics of the product without
being overly generic. Thus, these problems limit the usability of natural
language explanations (NLE) produced by existing explainable recommendation
models.

In order to address the issue that models commonly generate univiersally
correct explanations, previous works experimented with diversifying generated
reviews using distantly retrieved images as additional signals (Geng et al.
2022). However, recommender system datasets do not always have associated
images, and Geng et al. proposed to retrieve images from the web using
available textual data. While this method indeed significantly diversifies the
generated natural language explanations (NLE), there is no guarantee that the
retrieved content will truthfully represent the quality of the corresponding
product. Thus, the generator needs to condition on a given feature or aspect
the user cares about at inference time, commonly extracted from the ground-
truth review. This limits the usability of models as such user input might not
be available at inference time. Another line of work attempted to incorporate
a pre-trained language model for better generation quality (Li, Zhang, and
Chen 2021). However, the same study shows that pre-trained language models
such as GPT-2 struggle to produce diverse reviews while maintaining
competitive recommendation accuracy. Thus, generating informative and factual
reviews without having access to information in ground truth reviews remains
an open problem.

Figure 1: The proposed framework PRAG. Given the input user and item, our
model produces a rating estimation (Section 4.5), as well as a latent query
$Q$ (Section 4.4). We then use the query to retrieve existing reviews as
evidence for the recommendation, and probe a question answering model for
natural language explanation (Section 5).

Recent advances in knowledge-grounded NLG show that retrieving unstructured
text as supplements for the model’s parametric knowledge significantly
improves factuality and diversity of generated content (Lewis et al. 2020; Guu
et al. 2020). Inspired by such success, in this work, we propose to leverage
existing reviews as additional context for generating recommendation
explanations. Specifically, we propose Personalized Retriever Augmented
Generator (PRAG), a model loosely based on the retriever-reader framework for
explainable recommendation. PRAG consists of a personalized retriever that can
accurately give a rating estimation and generate a latent query for review
prediction given the input user and item. To encourage factual explanations,
we formulate the task of NLE generation as question-answering, where a reader
model produces the explanation grounded on retrieved content. Meanwhile, to
ensure the informativeness of the explanations, we estimate a set of
personalized high-tf-idf keywords using the latent query, and use these
keywords to guide the answer of the reader model. In this way, the reader
model produces the final explanation by abstracting input text snippets and
keywords, which yields superior quality compared to previous work in both
automatic and human evaluation. Our contributions are as follows:

  * •

To our best knowledge, we are the first work that evaluates and highlights the
importance of factuality in natural language explanation for recommender
systems.

  * •

We develop a novel personalized retriever-reader model for generating factual
and informative recommendation explanations.

  * •

We personalize question-answering for generating recommendation explanations.

##  2 Related Work

Explainable recommendation aims at providing users insight for a recommender
systems’ decision. Following earlier works that provide topic words and
product features as explanations (McAuley and Leskovec 2013; Zhang et al.
2014), recent works are increasingly focusing on generating reviews as
explanations (Ni et al. 2017; Ni and McAuley 2018; Hada, Vijaikumar, and
Shevade 2021; Dong et al. 2017). However, these works are not focused on the
factuality of the generated content. Meanwhile, existing works require
training a language model from scratch, while previous attempts at leveraging
pre-trained language models either require decoding-time search (Hada,
Vijaikumar, and Shevade 2021), or are not as performant as other
recommendation models (Li, Zhang, and Chen 2022).

It is to be noted that there are commonly used existing metrics such as
Distinct-N (Liu et al. 2022) and unique sentence ratio (USR) (Li, Zhang, and
Chen 2020) that focus on diversity of generated explanations. However, a
common measure used to increase diversity is to generate a sentence based on
words and phrases from the ground-truth review, such as in Geng et al. and Ni
and McAuley. Our work differs from previous works in that we consider
specifically the case where no information from the ground truth review is
given, which is common in recommender systems.

Outside of recommender systems, there is a general trend for using natural
language as explanations for various tasks, such as text classification
(Hancock et al. 2018; Zhou et al. 2020), image captioning (Majumder et al.
2022; Marasović et al. 2020) and question answering (Dalvi et al. 2021;
Tafjord and Clark 2021). However, expert annotated explanations are usually
unavailable in recommender system datasets, and reviews are usually noisy by
nature and require further processing (Bražinskas, Lapata, and Titov 2020,
2021). This leaves learning to generate explanations from noisy supervision an
open challenge, which we seek to address in this work.

Another related existing problem in natural language generation tasks is safe
and generic outputs. For example, this problem is well studied in dialogue
generation, where previous work shows that exposure bias and maximum
likelihood objective lead the model to produce universally applicable
responses (Zhou, Li, and Li 2021). In another parallel line of work on
evaluating natural language explanations, Wiegreffe, Marasović, and Smith
pointed out that universally correct explanations are undesirable, calling for
diverse and informative explanations. However, these works commonly focus on
traditional NLP tasks and well-formulated datasets.

##  3 PRAG: Setup and Overview

We first introduce the joint review-rating prediction task, then cover our
model’s general architecture.

###  3.1 Problem Setup

Consider a set of users $U$ and items $I$, where each user $u\in U$ and item
$i\in I$ in the training set is associated with a real numbered rating
$r_{u,i}\in\mathcal{T}$ and a review $e_{u,i}\in E$ that serves as
explanations for the users rating decision. The task of joint rating-
explanation estimation learns a function
$\operatorname{rec}:u,i\to\hat{r}_{u,i},\hat{e}_{u,i}$, where $\hat{e}_{u,i}$
is a textual explanation that informs the user of the reasoning behind
$\hat{r}_{u,i}$. Note that broadly speaking, explanations can be in various
forms, such as topic words and feature importance, here, we use $e_{u,i}$ to
denote natural language explanation specifically.

###  3.2 Model Overview

The architecture of PRAG is depicted in Figure 1. Given the input user $u$ and
item $i$, we first obtain semantic embeddings for all related existing
reviews. Here, “related reviews” are reviews written by the user or written
about the item. Then, given the historical reviews, a personalized retriever
model produces a latent query $Q_{u,i}$ that is close to the ground truth
explanation given the input user and item. A rating prediction module then
uses such a latent query to produce a rating estimation.

To generate an explanation, we exploit the rich semantic information in
$Q_{u,i}$. Specifically, we (1) retrieve a set of existing reviews $G$ based
on $Q_{u,i}$, where $G\subseteq E$, and (2) generate a set of high tf-idf
scored keywords that the user would care about based on $Q_{u,i}$ and $G$.
Finally, a keyword-guided question answering model produces the final
explanation based on the keywords, retrieved reviews, and the question. We now
provide descriptions for each component.

##  4 Personalized Retriever

The architecture of our retriever model is as shown in Figure 2. The review,
item, and user embedding go through a stack of transformer blocks and are then
pooled to generate the estimated review embedding.

Figure 2: Overview for the retriever architecture. Our model takes as input
user and item id and a set of corresponding historical reviews, then produces
a latent review $Q$. The latent review is then used for rating prediction,
retrieval, and explanation generation.

###  4.1 Embedding Reviews

Pre-trained language model representations contain rich semantic information
about sentences and supports arithmetic operations such as addition and dot
product. Following previous success in pre-computing embeddings for efficient
retrieval, such as in RAG (Lewis et al. 2020), we obtain sentence embeddings
for all reviews using a pre-trained sentence encoder $\mathrm{\Phi}$ that
outputs $d$ dimensional sentence representations. In practice, we adopt the
MPNET (Song et al. 2020) model from huggingface transformers (Wolf et al.
2020) for balanced efficiency and embedding quality. We note any state-of-the-
art general-purpose sentence encoding model can be used for PRAG, and provide
additional analysis with T5 (Raffel et al. 2020) sentence encoders in our
experiment section.

###  4.2 Review Aggregation

Our base personalized retriever model is based on BERT architecture (Devlin et
al. 2019). Specifically, we treat each historical review in the training
corpus as a token. When predicting the review a user would give to an item,
the input to the model is a set of historical reviews written by the user and
written for the item. Since these tokens do not represent consecutive words in
a sentence as in the original BERT model, we do not use any position
embedding. However, historical reviews play different roles when they are
related to the item versus being related to the user, thus we choose to
maintain learnable embedding $c\in\mathbb{R}^{d}$ to represent these two
distinct scenarios, which are added to the original review embedding instead
of position embeddings.

Finally, we maintain a learnable embedding $v_{u}\in\mathbb{R}^{d}$ and
$v_{i}\in\mathbb{R}^{d}$ for each user and item to model user preferences,
following common practice in recommender systems. For the input user and item,
we look up the corresponding embedding and add these embeddings to each of the
input review embedding. The final input review embedding to the model is thus:

$v_{u}+v_{i}+c.$

Such a final embedding is then passed through a stack of 2-layer transformers
to process the cross-review relationship in the input.

###  4.3 Personalized Attention

Not every review in the input is important. Thus, it is crucial to select
reviews that are helpful to the recommendation model. To achieve this, we
develop a personalized attention module for weighing review representations
from the base transformer model. Specifically, we obtain an attention score
with respect to each piece of input reviews using a standard linear layer with
a Rectified Linear Unit (ReLU) activation function using the concatenation of
the review embedding, user embedding, and item embedding. Following previous
works’ insight that recommendation models benefit from un-smoothed attention
scores’ ability to discard irrelevant items (Zhou et al. 2018), we conduct
weighted pooling directly using the normalized attention scores.

###  4.4 Review Embedding Prediction

Finally, the weighted sum of review embeddings is passed through a multi-
layer-perception (MLP) layer to produce the final latent query
$Q_{u,i}\in\mathbb{R}^{d}$. During training, we minimize the $\mathrm{L2}$
distance between the produced query and the embedding of the corresponding
ground truth review, $\mathcal{L}_{\text{retrieve}}$, following previous
success in using vectorized reviews to regularize recommender models (Hada,
Vijaikumar, and Shevade 2021). Such embedding thus represents the predicted
semantics of the input user and item, which could then be used to retrieve
relevant reviews that are semantically similar to the predicted review from
the existing review corpus.

###  4.5 Rating Prediction

To perform rating prediction, we combine HFT (McAuley and Leskovec 2013), a
strong matrix-factorization based explainable recommendation model with a
modified wide-and-deep (Cheng et al. 2016) architecture. Specifically, the
original HFT model makes predictions by modifying the following equation:

$\operatorname{rec}(u,i)=\gamma_{u}\times\gamma_{i}+\beta_{u}+\beta_{i}+\mu,$

where $\mu$ is set to the global mean value of all ratings, and
$\beta_{u},\beta_{i}$ are the learned bias. The model further ties either
$\gamma_{u}$ or $\gamma_{u}$ to topic models, and learns product or user-
specific topics by jointly minimizing the rating regression loss and the
negative likelihood of the corresponding review.

We extend the HFT model by using latent query $Q_{u,i}$. Specifically, we
adopt the estimated latent review as a new source of semantic information in
place of topic models. For example, when associating item features to semantic
information, we use a simple multi-layer-perception (MLP) to map $Q_{u,i}$ to
$\gamma_{i}$ (or $\gamma_{u}$). Meanwhile, previous work has shown that using
a shallow linear (wide) layer for memorizing simple patterns could increase
the performance of recommendation models (Cheng et al. 2016). Following this
intuition, we add an additional linear (wide) layer to the original model

$\operatorname{rec}(u,i)=\operatorname{MLP}(Q)\times\gamma_{i}+\operatorname{wide}(Q)+\beta_{u}+\beta_{i}+\mu.$

We learn to predict rating using the standard squared loss
$\mathcal{L}_{\text{rating}}$. The final joint loss for training the
personalized retriever is then

$\mathcal{L}_{\text{retrieve}}+\mathcal{L}_{\text{rating}}.$

##  5 Explanation Generation as Keyword-guided Question Answering

To generate an explanation, we source information from retrieved reviews and
treat the task of explanation generation as question answering. Specifically,
the reader model should be able to answer why a higher (or lower) rating
adjustment score $\gamma_{ui}$ is being produced, as this is the only inter-
user-item factor in rating prediction. To achieve this, we first train an
embedding estimator that generates informative keywords using the mean
embedding of latent query $Q_{u,i}$ as well as its corresponding set of
retrieved reviews. After this, we probe a question-answering model trained to
factually reflect content from the input with natural language prompts while
incorporating the generated keywords. The schema of the explanation generation
pipeline is as shown in Figure 3.

To facilitate factual behavior for both the embedding estimator and the
question-answering model, we design an aggregated task where the ground-truth
is guaranteed to have a strong correlation to model input at training time.
Concretely, we train a model to recover informative keywords in a review from
the latent vector encoded from that specific review. Now we discuss each of
the components in detail.

###  5.1 Retrieving Reviews

Given that the latent query is optimized to be similar to the ground truth
review, a natural scheme for review retrieval is to rank existing reviews’
semantic embedding with respect to $Q_{u,i}$ using similarity metrics.
However, in practice, we found the model often produces overly-generic
retrieval results, i.e., the reviews for multiple users tend to be similar, as
observed in previous work (Geng et al. 2022). To address this issue, we
propose to retrieve existing reviews using characteristics that are specific
to the user. In other words, we want to de-emphasize explanations that a model
will produce for every user based on the item. We do so by examining the
explanation for other users, estimating the latent explanation that are
produced for all users, and marginalizing out such a universal explanation.

Concretely, we sample a batch of users at inference time, and obtain each
sampled user’s corresponding latent query $Q_{{u_{n}},i}$. Then, we estimate
the explanation that would be produced for every user using the mean embedding
of the batch of predicted queries. Finally, we subtract such mean embedding
from the original $Q_{u,i}$, effectively marginalizing out the user-agnostic
aspect of the latent query. Since this changes the magnitude of $Q_{u,i}$, we
rank existing reviews using cosine distance instead of $\mathrm{L2}$ distance,
and select top reviews as the retrieved content. Note that if $Q_{u,i}$ is
tied to $\gamma_{i}$, we marginalize out the universal explanation for a batch
of items instead, as we need to de-emphasize the explanation being produced
for every item. We provide qualitative and quantitative analysis of such
marginalization in Section 8.

Figure 3:  PRAG use retrieved historical reviews as source text, and generates
the explanation based on the keywords.

###  5.2 Informative Keyword Generation

Retrieved reviews are by nature noisy, and multiple aspects in the reviews
could simultaneously contribute to the model’s rating decision. However, to
generate realistic reviews, the generator must select a few aspects to focus
on, as natural reviews typically focus on a few central topic words. To ensure
the informativeness of the generated review, we design an embedding estimator
that generates high tf-idf scored keywords given the personalized latent
query. We then use the keywords in the explanation generation phase for
improved informativeness.

To achieve this, we adopt a pre-trained GPT-2
model111https://huggingface.co/distilgpt2, and formulate the optimization
objective as the task of embedding-to-keyword prediction. Specifically, given
a latent embedding $\Phi(e)$ encoded from an original review $e$ in the
training set, we train the embedding estimator on recovering the 5-highest tf-
idf scored words from the original review. In practice, we use the
concatenated target keywords as desired model output and fine-tune the
language model using the standard MLE objective. At inference time, we could
then condition the embedding estimator on the corresponding latent query since
the query is optimized to be in the semantic space of the sentence encoder
$\Phi$.

###  5.3 Explanation Generation

For explanation generation, we adopt a general purpose question-answering
model222https://github.com/allenai/unifiedqa , and exploit the in-context
learning ability of large-scale, pre-trained language models (Brown et al.
2020). Specifically, we ask the model to answer the question “what was great?”
or “what was not good?”, depending on the sign of the predicted rating
adjustment score. Meanwhile, we note that the choice of prompt does not
significantly influence the generation quality, and provide experiment with
alternative prompts in the appendix.

To guide the model to cover the aspect that the user cares about, we want to
encourage the model’s output to contain at least one of the keywords produced
by the embedding estimator. To achieve this, we first apply constrained
decoding (Hokamp and Liu 2017) using the original, unguided question answering
model, forcing the output to include at least one of the input keywords.
However, this could result in ungrammatical text. To address this issue, we
manually rephrase 100 outputs for each dataset, which is easily achievable.
This results in a high-quality dataset where each input text is paired with a
set of keywords, with a ground-truth answer that contains at least one of the
keywords. We then fine-tune the original question answering model on this
dataset for a maximum of 10 epochs, following previous work on few-shot
question answering (Ram et al. 2021). The fine-tuned model could then be
applied for efficient explanation generation without any decoding-time
constraints.

##  6 Experiments

###  6.1 Datasets

We conduct our experiments on three publicly available dataset and splits from
various domains (Li, Zhang, and Chen 2020):
Yelp333https://www.yelp.com/dataset/challenge (restaurant reviews),
TripAdvisor444https://www.tripadvisor.com  (hotel) and Amazon Movies and TV
(He and McAuley 2016). Note that the data splits guarantee that products in
the test set always appear in the training set.

| Entail |  | D-1 |  | D-2 |  | ENTR |  | USR |  | MAUVE |
---|---|---|---|---|---|---|---|---|---|---|---|---
Method | Movie | Trip | Yelp |  | Movie | Trip | Yelp |  | Movie | Trip | Yelp |  | Movie | Trip | Yelp |  | Movie | Trip | Yelp |  | Movie | Trip | Yelp |
Att2Seq | 25.6 | 12.2 | 35.9 |  | 39.9 | 34.6 | 43.1 |  | 75.9 | 75.4 | 78.1 |  | 9.56 | 8.11 | 8.44 |  | 41.7 | 21.0 | 39.9 |  | 3.0 | 1.4 | 3.9 |
NRT | 36.1 | 10.0 | 31.4 |  | 44.0 | 32.4 | 41.0 |  | 77.8 | 72.8 | 76.6 |  | 7.5 | 7.5 | 8.3 |  | 36.1 | 46.3 | 44.4 |  | 3.0 | 3.0 | 4.2 |
PETER | 29.0 | 17.5 | 44.5 |  | 27.7 | 26.8 | 29.5 |  | 58.6 | 60.7 | 60.4 |  | 10.5 | 10.1 | 10.7 |  | 60.7 | 57.2 | 58.2 |  | 3.7 | 2.3 | 2.2 |
PEPLER | 17.9 | 11.0 | 16.0 |  | 23.2 | 23 | 25.5 |  | 51.5 | 52.2 | 52.5 |  | 11.1 | 10.0 | 11.0 |  | 52.6 | 41.7 | 49.1 |  | 1.1 | 0.4 | 0.4 |
Optimus | 25.1 | 22.8 | 11.5 |  | 31.9 | 32.8 | 33.2 |  | 77.3 | 77 | 79.3 |  | 10.3 | 8.5 | 10.7 |  | 98.5 | 92.1 | 96.1 |  | 3.5 | 3.3 | 4.5 |
SUM | 49 | 29.5 | 30.8 |  | 22.1 | 18.7 | 20 |  | 67.1 | 61 | 63.7 |  | 11.2 | 10.4 | 11.5 |  | 95.3 | 94.7 | 94.8 |  | 5.8 | 4.7 | 5.4 |
PETER+ | 40.0 | 32.6 | 59.4 |  | 43.9 | 42.6 | 47.0 |  | 78.4 | 81.9 | 83.1 |  | 9.48 | 8.53 | 9.85 |  | 60.6 | 31.5 | 52.8 |  | 12.9 | 5.3 | 10.4 |
PRAG | 88.8 | 80.1 | 86.2 |  | 45.6 | 39.9 | 47.1 |  | 84.3 | 82.2 | 84.7 |  | 12.0 | 12.0 | 11.9 |  | 71.8 | 76.5 | 70.4 |  | 23.1 | 42.8 | 20.3 |

Table 1: Automatic evaluation results on test sets. Entail denotes the
percentage of entailment among the generated reviews. D-1 and D-2 scores
denote Distinct-1 and Distinct-2 scores. ENTR denotes the geometric mean of
n-gram entropy among the generated text. USR denotes the ratio of unique
sentences generated. MAUVE is a distribution based-metric to measure the
quality of the generated text. Best and second-best performances are bolded
and underlined, respectively. See Section 6.3 for details.

###  6.2 Baselines

We compare four commonly used models in the literature: Att2Seq (Dong et al.
2017), NRT (Li et al. 2017), PETER (Li, Zhang, and Chen 2021), and PEPLER (Li,
Zhang, and Chen 2022). Among the models, Att2Seq and NRT are LSTM-based
models, while PETER is a transformer-based model. We additionally incorporate
a variant of PETER that conditions on a topic word from ground-truth review at
inference time, denoted by PETER+. Finally, PEPLER adopts a pre-trained GPT-2
model with prompt tuning for explanation generation. We provide more details
for each of the models in the appendix.

Further, we show that when augmented with our personalized retriever, opinion
aggregation approaches such as summarization models could seamlessly integrate
into the retriever-reader framework. We demonstrate this by proposing two
novel baselines, where summarization models are used to aggregate retrieved
reviews.

#### PRAG-Optimus (Optimus)

Adopts a state-of-the-art pre-trained VAE-based language model based on BERT
and GPT (Li et al. 2020). Recent studies find VAE-based language models could
be used for unsupervised opinion summarization (Iso et al. 2021). However, Iso
et al.’s approach requires searching over large amounts of potential sequences
at inference time, particularly when there are multiple inputs. Thus, we use
the original Optimus model without inference-time searching as our baseline.
Specifically, we fine-tune a pre-trained Optimus model on each dataset for 1
epoch on language modeling, per recommendation by the original authors, and
condition the GPT-based generator on the mean embedding of retrieved reviews
at inference time.

#### PRAG-SUM (SUM)

Following previous success in training summarizers by learning to recover the
target review from a set of distantly retrieved similar reviews (Amplayo and
Lapata 2020), we train our summarizer in a leave-one-out fashion over sets of
similar reviews. Specifically, for each review in the training set, we
retrieve a set of most similar reviews using cosine distance in sentence
encoder $\Phi$’s semantic space. These retrieved reviews then serve as the
input for the model. Then, the summarization module is trained to recover the
original review that is used for retrieval from the retrieved reviews. For
training, we use the same pre-trained T5 model as PRAG as our base model for a
fair comparison and train the model using maximum likelihood estimation (MLE)
objective. We provide more details for data processing and training of SUM in
the appendix.

###  6.3 Automatic Evaluation

For automatic evaluation, we generate 10,000 samples from each baseline model,
and measure the performance in terms of factuality, informativeness, and
generation quality. We provide additional recommendation performance analysis
on all eligible models on the whole test set.

#### Factuality.

Factuality constraints that the models’ generated explanations are factually
correct. Intuitively, the statement in a generated explanation is correct if
it could be supported by any existing reviews in the training data. Otherwise,
the model is exhibiting hallucinations (Maynez et al. 2020). As reported by
the same work, entailment models could better measure the factual consistency
between reference text and generated content. We follow their findings and
evaluate the entailment relationship between generated reviews with reviews of
the same product in the training set. Specifically, for each generated
explanation, we check whether the explanation entails any reviews from the
training set for the same product using a pre-trained entailment model
555https://huggingface.co/prajjwal1/roberta-large-mnli. We note that a
generated explanation is factual if it entails any piece of existing reviews
for the product, and report the entailment ratio. Specifically, we report the
percentage of entailment as Entail.

#### Informativeness.

We measure the informativeness of generated explanations using token,
sentence, and corpus level evaluations. Concretely, We evaluate the models
using Distinct-1 and Distinct-2 (D-1, D-2) scores (Li et al. 2016), Unique
Sentence Ratio (USR) as proposed by (Li, Zhang, and Chen 2020) and ENTR
(Jhamtani et al. 2018) following previous work on diversifying generated
content (Majumder et al. 2021).

#### Generation quality.

To measure the generated explanation quality, we opt to measure how human-like
the generated explanations are. Specifically, we adopt MAUVE (Pillutla et al.
2021), a distribution-based evaluation metric that measures how close the
generated contents are to the ground truth corpus.

#### Recommendation performance.

We evaluate the recommendation performance using Rooted Mean Squared Error
(RMSE) score following previous works done using the same dataset.

##  7 Results

###  7.1 Quantitative Evaluation

We report PRAG’s performance compared to baseline models as in Table 1. PRAG
consistently outperforms the baseline models in terms of both diversity and
informativeness, and can generate high-quality sentences compared to human-
written reviews.

#### Retrieval component improves generation quality.

As shown in Table 1, conditioning on additional information improves both
model diversity and factuality. Specifically, Optimus, SUM, and PRAG
consistently outperform other models that only maintain vector representation
for each user and item. The generated content has both higher sentence-wise
diversity (from D-1 and D-2 scores) and higher corpus-level diversity (from
ENTR and USR scores), as well as being closer to human written reviews (from
MAUVE scores). This shows that the personalized retrieval component could
improve generation quality.

#### Training data affects generator factuality.

While retrieved historical reviews encourage more factual output in general,
this does not guarantee strong factuality. In particular, the T5-based
summarizer and Optimus model have limited improvement in terms of factuality
compared to other baseline models. This highlights the major cause of
hallucination is directly training generators on noisy data. Specifically,
Maynez et al. reported that ground-truth sequences in the training data that
contain hallucinated content would trigger the model to be less factual at
inference time in summarization. In parallel, Longpre et al. reported that
noisy retrieved content would cause the generator to hallucinate for question
answering.

We note that a similar case applies to review generation as well. In
particular, there is no guarantee that the ground truth review will entail any
of other reviews for the same product in the training set. While this behavior
is natural for human (Maynez et al. 2020), the model ended up learning to
hallucinate during optimization as a result. By leveraging a reader model
trained to faithfully present content in the input text, PRAG sidesteps this
issue, and thus has the best factuality performance across models.

###  7.2 Recommendation Performance

| Movie | TripAdivor | Yelp
---|---|---|---
NRT | 0.79 | 0.95 | 1.01
PETER | 0.80 | 0.95 | 1.01
PEPLER | 1.71 | 1.25 | 1.69
PRAGu | 0.80 | 0.95 | 1.01
PRAGT5 | 0.80 | 0.96 | 1.00
PRAG | 0.79 | 0.95 | 1.01
Table 2: RMSE scores for recommendation performance. By default, PRAG uses
MPNET sentence embedding and ties $Q_{u,i}$ to $\gamma_{i}$. PRAG T5 uses
embedding from T5 model and PRAG(u) maps $Q_{u,i}$ to $\gamma_{u}$.

To verify PRAG’s ability for recommendation, we report RMSE scores on the
three datasets as shown in Table 2. Since SUM and Optimus are based on the
same retriever model as PRAG, we report PRAG’s performance only in this
section. Further, Att2Seq cannot produce rating estimations, and thus we omit
the model in the table. We validate that the performance of baseline models is
consistent with previous works on the same dataset (Geng et al. 2022). As
shown in Table 2, PRAG achieved state-of-the-art rating estimation
performance.

###  7.3 Human Evaluation

| Fluency |  | Informativeness |
---|---|---|---|---
PRAG vs. | Movie | Trip | Yelp |  | Movie | Trip | Yelp |
NRT | 37 | 16 | 13 |  | 28 | 1 | 3 |
PETER | 20 | 26 | 15 |  | -2 | 12 | 27 |
SUM | 8 | 5 | 3 |  | 0 | -6 | 5 |

Table 3: Human evaluation results of PRAG versus baseline models. The number
reported are the number of additional times PRAG win against the opponent
model. Note both PRAG and SUM are retriever augmented generators.

We perform human evaluation using 150 test samples with several of the
strongest baselines. Specifically, we compare PRAG’s performance against NRT,
PETER and SUM. We omit Att2Seq and PEPLER since the models are not as
performant as other models in recommendation performance, and leave PETER+ out
since ground-truth aspects are not always available. Finally, we pick the
summarizer as a stronger retriever-augmented baseline (compared to Optimus)
based on automatic evaluation results. We compare the generated reviews in
terms of (1) fluency compared to other models (2) informativeness to the user.
We report the results in Table 3. Similar to automatic evaluation results, the
retriever could reliably boost generation performance: PRAG is consistently
ranked as the most fluent model by human evaluators, while SUM has almost on-
par performance due to having access to the personalized retriever. Meanwhile,
PRAG and SUM almost invariably win in terms of informativeness compared to
previous works.

##  8 Analysis and Discussion

#### Analysis of marginalization and retrieval reliability.

To validate whether marginalizing the predicted embedding before performing
retrieval truly uncovers user or item characteristics instead of resulting in
random vectors, we perform a simple test by comparing the retrieval result
between a pair of mirroring retrievers. Specifically, we hypothesize that
given a user and an item, a retriever that ties $Q_{u,i}$ to $\gamma_{u}$ and
a retriever that ties $Q_{u,i}$ to $\gamma_{i}$ should achieve agreement after
marginalization. That is, the two retrievers should be able to retrieve
similar reviews. To verify this, we report the average exact-match between two
retrievers using agreement at 5. As shown in Table 5, the retrievers
consistently achieved significantly higher agreement compared to a random
baseline across three datasets. Further, we report examples of retrieved
reviews with and without marginalization. As shown in Table 4, after
marginalization, there is a clear trend that reviews related to aesthetic
aspects are being retrieved, as opposed to a set of generic reviews without
marginalization.

#### Analysis of sentence embedding model.

We hypothesize that any strong sentence embedding model could be used for
PRAG. To validate such a hypothesis, we train PRAG model on all three datasets
using a pre-trained T5-based sentence encoder. We report the performance as in
Table 2. As shown, there is no significant performance variation across
different types of sentence embeddings.

Retrieved Reviews
---
the decor of the hotel is greatly refined. … the building and decoration are
very nice and very tastefully decorated. a four seasons in every respect it is
an architecturally interesting and esthetically pleasing property in a great
location. the rooms are stunning.
Table 4: Retrieved reviews (cropped, from top-5 results). With marginalization, the retrieved reviews consistently focus a specific aspect. See appendix for comparison with non-marginalized retrieval results.  | Movie | TripAdivor | Yelp
---|---|---|---
Retriever | 2.87 | 2.44 | 2.50
Random | 0.54 | 0.61 | 0.27
Table 5: Average agreement-at-5 of mirroring retrievers.

#### Effect of tying user or item Factors to latent query.

Similar to the HFT (McAuley and Leskovec 2013) model, the rating estimation
component in PRAG could either tie the user or item factor to $Q_{u,i}$. We
conduct experiments using both types of architectures, and also report our
results in Table 2. Similar to findings reported for the HFT model (McAuley
and Leskovec 2013), the performance is generally dataset-dependent, and the
design could be viewed as a hyper-parameter.

##  9 Summary and Outlook

In this work, we propose PRAG, a retriever-reader model that can generate
factual and diverse explanations for recommendation. Experiments on three
real-world datasets show PRAG can generate both factually grounded and
informative explanations. We also investigate the cause of hallucinated
content in review generation, and demonstrate the benefit of training text
generation models on hallucination-free tasks and datasets. Meanwhile,
although we adopted a question-answering model for explanation generation,
PRAG’s retrieval component could provide support for personalizing a wider
range of knowledge-based tasks, such as personalized conversational
recommendation, summarization, and product description generation.

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compat=1.1.0

# Extended Scotogenic Model of Neutrino Mass and Proton Decay

Takaaki Nomura  0000-0002-0864-8333<EMAIL_ADDRESS>College of Physics,
Sichuan University, Chengdu 610065, China  Oleg Popov  0000-0002-0249-8493
<EMAIL_ADDRESS>(corresponding author) Faculty of Physics and Mathematics and
Faculty of Biology, Shenzhen MSU-BIT University,
1, International University Park Road, Shenzhen 518172, China

###### Abstract

The current article presents an extension of the classical scotogenic neutrino
mass paradigm, where the three issues in particle physics: dark matter,
smallness of neutrino mass, and stability of the proton are interconnected.
The scenario encompasses the neutrino mass as well as the proton decay as a
consequence of an existence of the dark matter. The study successfully
achieves the correlation between the naturally small neutrino masses and
naturally long proton lifetime in the present paradigm. Furthermore, all
relevant cosmological, collider, and flavor physics constraints are
incorporated in the detailed analysis. The scotogenic fermionic dark matter
with the mass in the range from $100$ GeV to $10$ TeV successfully satisfies
all relevant constraints. The valid range of $2.51\times
10^{-5}<\lambda<2\times 10^{-3}$ is obtained for the 2HDM $\lambda$ coupling.
We give a brief discussion, as well as, outline some of the future prospects.

neutrino mass, proton decay, proton stability, dark matter

††preprint: 2312.xxxx

##  1 Introduction

T he origin of neutrino mass is a challenging and long standing problem in
particle physics. First thought to be massless, experimental data of observed
solar and atmospheric neutrinos from the late 90’s changed physicists approach
on the problem [1, 2]. It had been already known, at the time, neutrino
oscillation is the one possible phenomena to explain the lack of the solar
anti-electron neutrinos [3, 4] and anomaly in the atmospheric anti-muon
neutrinos [5]. Neutrino oscillations require tiny neutrino masses,
indistinguishable by the detectors, and lepton mixing that is described by
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [6, 7, 8]. As a result
oscillation data fixes the neutrino mass splittings and mixing angles of the
lepton sector [9]. In addition physicists already had developed mechanisms
that would explain the tiny neutrino masses [10, 11, 12, 13]. Starting with
tree level [10, 11, 12], then one loop order [13, 14, 15], and later
engineering more various models [16] challenging to naturally suppress
neutrino mass. The challenge still stands today and we still attempt to come
up with naturally suppressed pathways to small neutrino masses [14, 17] and
connect, also test, this models via various endless physics, including quark
sector, modular symmetries, phase transitions and gravitational waves,
topological effects, axions, and other interesting physics.

The other long standing problem in particle physics that remains unresolved
since 1970’s [18, 19, 20] is the stability of the proton [21, 22, 23, 24].
Since the inception of the grand unified theories (GUTs) [25] and the road to
unify all known internal symmetries in nature, the prediction of the proton
decay has been a smoking gun signature of any unification scenario. After 70
years of search in the chase of the proton decay signal, the proton decay
still remains to be observed. Many experiments [26, 27, 28, 29] over the globe
build larger detectors and look for more exotic proton decay channels [30, 31,
32] in hope to reach the Holy Grail. Several GUT models have since been ruled
out, for instance, minimal $SU(5)$ [33]. This situation motivated model
builders to put forward models with naturally suppressed proton decay, some
examples are: $SU(5)$ based models [34], models with exotic proton decay
channels [35, 21, 22, 36], radiatively generated proton decays [21, 22, 23],
and others [23]. As it turned out, creating a GUT model with naturally
suppressed proton decay is a no man’s land territory. One has to find a smart
way: including adding more symmetries, irreducable representations, and so on;
in order to make the proton decay width naturally small or even obtain a
completely stable proton.

Under the light of everything discussed above, in the present manuscript we
try to put forward the minimal scenario where two aforementioned problems in
particle physics are united. We will present a scenario where neutrino mass
and proton decay width are naturally suppressed and are correlated. Both
achievements require a lepton number violation, hence neutrinos are majorana
in nature. Furthermore, both the neutrino mass and proton decay width vanish
in the limit when lepton number is conserved. This creates a unique
opportunity to complement the neutrino data and the proton decay experimental
constraints together and test the enhanced scotogenic neutrino scenario. Scoto
from Greek meaning darkness [15]. As a bonus, the model also predicts a stable
dark matter.

The paper is organized as follows: Sec. 2 describes the model and its content,
neutrino mass generation is given in Sec. 3, proton decay and relevant details
are studied in Sec. 4, constraints on the model’s parameters are presented in
Sec. 5, results are given in Sec. 6, Sec. 7 contains the discussion, and Sec.
8 concludes the paper.

##  2 Model

The model presented here is minimal that is capable of generating neutrino
mass and proton decay from the common origin, which makes the two correlated.
In order to build the model that links the proton decay to the neutrino mass
one must go beyond the conventional seesaw scenarios and use fields that
simultaneously contribute to the neutrino mass generation and proton decay
width. Here, we present a model that adds _ad hoc_ $\mathbb{Z}_{2}$ symmetry,
similar to the scotogenic model [15], which makes our neutrino mass and proton
decay radiative and allows for the dark matter candidates to be present. From
one point of view, the model can be viewed as enhanced scotogenic model which
extends beyond just neutrino mass generation and encompasses the proton decay
width generation as well. At the end, we get a model where the existence of
the dark matter is linked to the smallness of the neutrino mass as well as to
the smallness of the proton decay width (_i.e._ proton’s life time), where the
last two are correlated.

The enhanced scotogenic model contains the Standard Model (SM) fields as well
as the basic scotogenic fields; Three generations of singlet fermion $N$ and
$SU(2)_{L}$ doublet scalar $\eta$ that are odd under the dark
$\mathbb{Z}_{2}$, while SM fields are even. Furthermore, three generations of
the vector-like $D$ electroweak singlet quark and $\tilde{R}_{2D}$, both odd
under the dark $\mathbb{Z}_{2}$, are part of the enhanced scotogenic model and
contribute to the radiative generation of the proton decay width. All fields
contained in the model and their transformations under the model’s symmetries
are given in Tab. 1.

Some important properties of the enhanced scotogenic model are both neutrino
mass and proton decay width generation is at loop order which contributes to
their smallness. Both, the neutrino mass and proton decay width, are
proportional to the source of the lepton number violation ($m_{N}$) and both
vanish when the last one approaches zero: no lepton number violation leads to
massless neutrinos and stable proton. The details of this correlation are
given in the later section of the manuscript.

Fields | $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $\mathbb{Z}_{2}$
---|---|---|---|---
$Q$ | $\pmb{3}$ | $\pmb{2}$ | $~{}\frac{1}{6}$ | $+$
$\bar{u}$ | $\pmb{\bar{3}}$ | $\pmb{1}$ | $-\frac{2}{3}$ | $+$
$\bar{d}$ | $\pmb{\bar{3}}$ | $\pmb{1}$ | $~{}\frac{1}{3}$ | $+$
$L$ | $\pmb{1}$ | $\pmb{2}$ | $-\frac{1}{2}$ | $+$
$\bar{e}$ | $\pmb{1}$ | $\pmb{1}$ | $1$ | $+$
$N$ | $\pmb{1}$ | $\pmb{1}$ | $0$ | $-$
$D,\bar{D}^{\dagger}$ | $\pmb{3}$ | $\pmb{1}$ | $-\frac{1}{3}$ | $-$
$H$ | $\pmb{1}$ | $\pmb{2}$ | $~{}\frac{1}{2}$ | $+$
$\tilde{R}_{2D}$ | $\pmb{3}$ | $\pmb{2}$ | $~{}\frac{1}{6}$ | $-$
$\eta$ | $\pmb{1}$ | $\pmb{2}$ | $-\frac{1}{2}$ | $-$
Table 1: Model field content.

Lagrangian for the model fields content (Tab. 1) is as follows

$\displaystyle-\mathcal{L}^{\text{Yuk}}_{\text{BSM}}$
$\displaystyle=Y_{1}\bar{D}L\tilde{R}_{2D}+Y_{2}QN\tilde{R}_{2D}^{\dagger}+Y_{3}QD\tilde{R}_{2D}$
(1a) $\displaystyle+Y_{4}LN\eta^{\dagger}+Y_{5}Q\bar{D}\eta$
$\displaystyle+m_{N}NN+m_{D}D\bar{D}+\text{h.c.}$ $\displaystyle V$
$\displaystyle=\mu_{H}^{2}\left(H^{\dagger}H\right)+\mu_{\eta}^{2}\left(\eta^{\dagger}\eta\right)+\mu_{R}^{2}\left(\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\right)$
$\displaystyle+\lambda_{H}\left(H^{\dagger}H\right)^{2}+\lambda_{\eta}\left(\eta^{\dagger}\eta\right)^{2}+\lambda_{R}\left(\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\right)^{2}$
$\displaystyle+\lambda_{R}^{\prime}\left(\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\right)+\lambda_{H\eta}\left(H^{\dagger}H\right)\left(\eta^{\dagger}\eta\right)$
$\displaystyle+\lambda_{H\eta}^{\prime}\left(H^{\dagger}\eta\right)\left(\eta^{\dagger}H\right)+\lambda_{HR}\left(H^{\dagger}H\right)\left(\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\right)$
$\displaystyle+\lambda_{HR}^{\prime}\left(H^{\dagger}\tilde{R}_{2D}\right)\left(\tilde{R}_{2D}^{\dagger}H\right)$
$\displaystyle+\lambda_{\eta
R}\left(\eta^{\dagger}\eta\right)\left(\tilde{R}_{2D}^{\dagger}\tilde{R}_{2D}\right)+\lambda_{\eta
R}^{\prime}\left(\eta^{\dagger}\tilde{R}_{2D}\right)\left(\tilde{R}_{2D}^{\dagger}\eta\right)$
$\displaystyle+\left(\lambda
HH\eta\eta+\lambda_{3R}\tilde{R}_{2D}\tilde{R}_{2D}\tilde{R}_{2D}\eta+\text{h.c.}\right)$
(1b)

The inert scalar doublet $\eta$ is written by

$\eta=\begin{pmatrix}\frac{1}{\sqrt{2}}(\eta_{R}+\imath\eta_{I})\\\
\eta^{-}\end{pmatrix},$ (2)

where $\eta_{R}^{-}$ has electric charge $-1$ and the components are mass
eigenstates since it does not mix with the SM Higgs.

Here we consider masses of scalar fields from inert doublet $\eta$. After
electroweak symmetry breaking, we obtain squared masses of inert scalar fields
as follows.

$\displaystyle m_{\eta_{R[I]}}^{2}$
$\displaystyle=\mu^{2}_{\eta}+\frac{v^{2}}{2}(\lambda_{H\eta}+\lambda^{\prime}_{H\eta}+[-]2\lambda),$
$\displaystyle m_{\eta^{\pm}}^{2}$
$\displaystyle=\mu^{2}_{\eta}+\frac{v^{2}\lambda_{H\eta}}{2},$ (3)

We thus find mass difference between $\eta_{R}$ and $\eta_{I}$ is given by
$m_{\eta_{R}}^{2}-m_{\eta_{I}}^{2}=2\lambda v^{2}$.

Potential minimization condition and SM doublet Higgs mass are given as in the
SM

$\displaystyle 0$ $\displaystyle=\mu_{H}^{2}+\lambda_{H}v^{2},$ (4a)
$\displaystyle m_{h_{R}}^{2}$ $\displaystyle=\mu_{H}^{2}+3\lambda_{H}v^{2},$
(4b) $\displaystyle m_{h_{I}}^{2}$ $\displaystyle=m_{h^{\pm}}^{2}=0,$ (4c)

where $\left\langle H\right\rangle=v/\sqrt{2}$. Also dark leptoquark mass is
given as
$m_{\tilde{R}_{2D}}^{2}=2\mu_{R}^{2}+\left(\lambda_{HR}+\lambda_{HR}^{\prime}\right)v^{2}$,
assuming $m_{R}^{2}\gg v^{2}$, and $\lambda_{HR}^{(\prime)}<1$ is taken.

Regarding BSM quarks, $N$ generates its majorana mass via lepton number
violating term $m_{N}NN$ (1a) and the mass of the BSM vector-like singlet dark
quark, $D$, is $M_{D}$ (1a). Model contains no BSM gauge bosons or vector
particles, while SM gauge bosons obtains their mass via SM Higgs as in the SM
scenario.

##  3 Neutrino Mass

Figure 1: Scotogenic neutrino mass.

In the model neutrno mass is radiatively generated by the scotogenic scenario
where the relevant one-loop diagram is shown in Fig. 1. Calculating the
diagram we obtain active neutrino mass matrix such that

$\displaystyle(m_{\nu})_{ij}=\frac{1}{32\pi^{2}}\sum_{k=1}^{3}Y_{4}^{ik}m_{N_{k}}Y_{4}^{jk}$
$\displaystyle\times\left(\frac{m^{2}_{\eta_{R}}}{m_{N_{k}}^{2}-m^{2}_{\eta_{R}}}\ln\frac{m^{2}_{\eta_{R}}}{m^{2}_{N_{k}}}-\frac{m^{2}_{\eta_{I}}}{m_{N_{k}}^{2}-m^{2}_{\eta_{I}}}\ln\frac{m^{2}_{\eta_{I}}}{m^{2}_{N_{k}}}\right),$
(5)

where $M_{N_{k}}$ is a mass eigenvalue of singlet fermion $N$. If the scale of
inert scalar boson mass is much larger than that of $N$, we can approximate
$m_{\nu}$ by

$(m_{\nu})_{ij}\simeq-\frac{1}{16\pi^{2}}\frac{\lambda
v^{2}}{M_{\eta}^{2}}\sum_{k=1}^{3}Y_{4}^{ik}m_{N_{k}}Y_{4}^{jk},$ (6)

where $M_{\eta}^{2}=(m^{2}_{\eta_{R}}+m^{2}_{\eta_{I}})/2$. In this form we
explicitly see that active neutrino masses vanish for $m_{N_{k}}\to 0$.

##  4 Proton Decay

The proton decay is induced at one-loop order through the ad hoc
$\mathbb{Z}_{2}$ symmetry, similar to the scotogenic scenario. We achieve the
radiative proton decay via introducing minimal BSM particle content and as a
result get a viable dark matter candidates, which are stabilized by the
scotogenic $\mathbb{Z}_{2}$ symmetry. Since the radiative proton decay
resembles the scotogenic structure for the scotogenic neutrino mass scenario,
it can be called an enhanced scotogenic mechanism. The radiative nature of the
proton decay origin augments the suppression of the width of the long lived
proton. Generating the scotogenic proton decay mechanism requires a minimal
content of $N$ electrically neutral dark fermion, $\tilde{R}_{2D}$ dark
leptoquark and BSM vectorlike $D$ electroweak (EW) siglet quark. Paricle
content is given in Tab. 1. Furthermore, due to scotogenic nature of the
mechanism, both neutrino mass and proton decay width are proportional to the
$N$ fermion’s majorana mass. This induces the correlation between the neutrino
mass origin and proton decay width generation. Both neutrino mass and proton
decay width posses the following characteristics: radiatively suppressed, both
are proportional to the lepton number violating term $m_{N}$ and vanish as
$m_{N}$ approaches zero, both get suppressed as the propagator’s mass $m_{N}$
becomes very large. As a result of the model construction, both neutrino mass
and proton decay width are naturally suppressed and are correlated, which
creates the opportunity to complement the experimental results from neutrino
mass, neutrino oscillations, and proton decay search data to test the enhanced
scotogenic model and make further predictions. The radiative proton decay
diagram is given in Fig. 2. The tree level diagrams are absent due to the
implementation of the scotogenic $\mathbb{Z}_{2}$ symmetry. Feynman diagram in
Fig. 2 contributing to proton decay width and scotogenic neutrino mass feynman
diagram in Fig. 1 are the leading feynman diagrams generating a correlated
scotogenic neutrino mass and proton decay in the framework of an enhanced
scotogenic scenario.

Figure 2: Feynman diagram contributing to proton decay.

In the model at hand proton has two dominant, two body, decay channels, Proton
decay via $p\rightarrow e^{+}\pi^{0}$ and $p\rightarrow\nu^{\dagger}\pi^{+}$.
The corresponding one loop induced proton decay widths are given as follows

$\displaystyle\Gamma\left(p\rightarrow
e^{+}\pi^{0},\nu_{\alpha}^{\dagger}\pi^{+}\right)$
$\displaystyle=\frac{m_{p}}{32\pi}\left(1-\frac{m_{\pi}^{2}}{m_{p}^{2}}\right)^{2}\left|Y_{2}^{1a}\frac{m_{N}^{ab}}{m_{\tilde{R}_{2D}}}Y_{2}^{b1}Y_{3}^{1c}\frac{m_{D}^{cd}}{m_{\tilde{R}_{2D}}}Y_{1}^{d\alpha}\frac{W_{0}^{l}\left(p\rightarrow
e^{+}\pi^{0},\nu^{\dagger}\pi^{+}\right)}{m_{\tilde{R}_{2D}}^{2}}\right.$ (7a)
$\displaystyle\times\left.\left[F_{0}(x_{N},x_{D},y)+F_{1}(x_{N},x_{D},y)\frac{q^{2}}{m_{\tilde{R}_{2D}}^{2}}\right]\right|^{2},$

where $\alpha=1$ for $p\rightarrow e^{+}\pi^{0}$, since the constraints on the
$p\rightarrow e^{+}\pi^{0}$ are more stringent compared to
$p\rightarrow\mu^{+}\pi^{0}$. On the other hand, for neutrinos situation is
different. So, $\alpha=1,2,3$ for $p\rightarrow\nu_{\alpha}^{\dagger}\pi^{+}$,
since the neutrino escapes and can be any flavour, while experimental
constraint applies to $p\rightarrow\nu^{\left(\dagger\right)}\pi^{+}$. Loop
functions, Kallen $\lambda$, and other definitions are as given below

$\displaystyle F_{0}(x_{N},x_{D},y)=\left\\{(1+x_{D})(x_{N}-1)y\ln
x_{D}-(x_{D}-1)\left[(x_{D}-x_{N})(x_{N}-1)\ln\left(\frac{x_{D}}{x_{N}}\right)-(1+x_{N})y\ln
x_{N}\right]\right.$ (8a)
$\displaystyle\left.-(x_{D}-1)(x_{N}-1)\lambda^{1/2}(x_{D},x_{N},y)\left[\ln(4x_{D}x_{N})-2\ln\left(x_{D}+x_{N}-y+\lambda^{1/2}(x_{D},x_{N},y)\right)\right]\right\\}$
$\displaystyle\times\left[2y(x_{D}-1)(x_{N}-1)\left((x_{D}-1)(x_{N}-1)+y\right)\right]^{-1},$
$\displaystyle
F_{1}(x_{N},x_{D},y)=\left\\{4y\left[(x_{D}-x_{N})^{2}+y\frac{1-6x_{D}x_{N}+2x_{N}^{2}+2x_{D}^{2}+x_{D}^{2}x_{N}^{2}}{(x_{D}-1)(x_{N}-1)}+y^{2}\frac{1-4x_{D}x_{N}+x_{D}^{2}+x_{N}^{2}+x_{D}^{2}x_{N}^{2}}{(x_{D}-1)^{2}(x_{N}-1)^{2}}\right]\right.$
$\displaystyle+\left(2\left[(x_{D}-1)^{2}(x_{D}-x_{N})^{2}-2y(x_{D}-1)\left(-x_{N}+x_{D}(-2+x_{D}+2x_{N})\right)+y^{2}\left(1-4x_{D}+x_{D}^{2}\right)\right]\right.$
$\displaystyle\left.\times\left[-x_{N}+y+x_{D}(1-x_{D}+x_{N}+y)\right]\ln
x_{D}(x_{D}-1)^{-3}+\left(x_{D}\leftrightarrow x_{N}\right)\right)$
$\displaystyle\left.-2\lambda^{3/2}(x_{D},x_{N},y)\left[\ln(4x_{D}x_{N})-2\ln\left(x_{D}+x_{N}-y+\lambda^{1/2}(x_{D},x_{N},y)\right)\right]\right\\}\frac{1}{24y^{2}\left[(x_{D}-1)(x_{N}-1)+y\right]^{2}},$
(8b) $\displaystyle\hskip
142.26378pt\lambda\left(a,b,c\right)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc,$ (8c)
$\displaystyle\hskip
142.26378ptx_{N}=\frac{m_{N}^{2}}{m_{\tilde{R}_{2D}}^{2}},\quad
x_{D}=\frac{m_{D}^{2}}{m_{\tilde{R}_{2D}}^{2}},\quad
y=\frac{m_{p}^{2}}{m_{\tilde{R}_{2D}}^{2}}.$ (8d)

Everything defined in dimensionless units for convenience of the reader.
Proton decay matrix elements and exchanged momentum between the quarks in the
proton are obtained from lattice calculations [37] and are given below

$\displaystyle W_{0}^{e}(p\rightarrow e^{+}\pi^{0})$
$\displaystyle=\left\langle\pi^{0}\right|(ud)_{L}u_{L}\left|p\right\rangle$
$\displaystyle=0.134(5)(16)~{}\text{GeV}^{2}\text{~{}\cite[cite]{[\@@bibref{Number}{Aoki:2017puj}{}{}]}},$
(9a) $\displaystyle W_{0}^{\nu}(p\rightarrow\nu^{\dagger}\pi^{+})$
$\displaystyle=\left\langle\pi^{+}\right|(du)_{L}d_{L}\left|p\right\rangle$
$\displaystyle=0.189(6)(22)~{}\text{GeV}^{2}\text{~{}\cite[cite]{[\@@bibref{Number}{Aoki:2017puj}{}{}]}},$
(9b) $\displaystyle q^{2}$
$\displaystyle=0.2~{}\text{GeV}^{2}\text{~{}\cite[cite]{[\@@bibref{Number}{Aoki:2017puj}{}{}]}}.$
(9c)

The $W_{1}^{l}$ for $l=e^{+}$ or $l=\nu_{\alpha}^{\dagger}$ are negligible
compared to $W_{0}^{l}$ and will be dropped in the further analysis [37].

##  5 Constraints

In this section we summarize relevant observables to constrain the model and
corresponding parameters which are limited by them.

The neutrino mass matrix is described by the parameters
$\\{\lambda,Y_{4},m_{\eta},m_{N}\\}$ as Eq. (6) where
$m_{\eta}=m_{\eta_{R}}\simeq m_{\eta_{I}}\simeq m_{\eta^{\pm}}$. Here we do
not reproduce neutrino mixings and require order of $m_{\nu}$ to be 0.1 eV in
order to obtain allowed magnitude of the parameters assuming no hierarchical
structure of $Y_{4}$.

The most stringent proton lifetime constraint for two body decay was obtained
by Super-Kamiokande (SK) collaboration and is given by $\tau(p\rightarrow
e^{+}(\mu^{+})\pi^{0})>2.4(1.6)\times 10^{34}$ years [38] and
$\tau(p\rightarrow\mu^{+}K^{0})>3.6\times 10^{33}$ years [39]. Furthermore,
the proton decay to any number of final states bound obtained from baryon
conservation limit is given by $\tau(p\rightarrow\text{any})\gtrsim 2\times
10^{30}$ years [40]. These constraints apply to the proton decay obtained for
our model and that was given in Sec. 4. The corresponding results of these
constraints are included in the Sec. 6.

In addition to the constraint from proton decay lifetime, we also consider
lepton flavor violations (LFVs) in the scotogenic model. As the most stringent
constraint we focus on $\ell\to\ell^{\prime}\gamma$ process in this work. The
branching ratio(BR) of the process is given by [41]

$\displaystyle\rm{Br}(\ell\to\ell^{\prime}\gamma)=$
$\displaystyle\frac{3(4\pi)^{3}}{4G_{F}^{2}}|(A_{D})_{\ell\ell^{\prime}}|^{2}{\rm
Br}(\ell\to\ell^{\prime}\nu_{\ell}\bar{\nu}_{\ell^{\prime}}),$ (10)
$\displaystyle(A_{D})_{\ell\ell^{\prime}}=$
$\displaystyle\sum_{i=1}^{3}\frac{(Y_{4}^{*})^{i\ell^{\prime}}Y_{4}^{i\ell}}{2(4\pi)^{2}}\frac{1}{m_{\eta^{\pm}}^{2}}F_{2}\left(\frac{M_{N_{i}}^{2}}{m_{\eta^{\pm}}^{2}}\right),$
(11)

where $F_{2}(x)$ is a function obtained from loop momentum integration:

$F_{2}(x)=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\ln x}{6(1-x)^{4}}.$ (12)

In our analysis, we consider elements of Yukawa coupling $Y_{4}$ have similar
size, and apply the strongest bounds from $\mu\to e\gamma$ [42] to estimate
typical allowed scale of masses $\\{m_{N_{i}},m_{\eta^{\pm}}\\}$ and the
magnitude of $Y_{4}$. We thus adapt

${\rm Br}(\mu\to e\gamma)<4.2\times 10^{-13}.$ (13)

We also take into account a constraint on the leptoquark mass
$m_{\tilde{R}_{2D}}$ from collider experiments. Firstly, it should be noted
that the leptoquark in the model has $Z_{2}$ odd parity and its decay mode is
different from those of typical leptoquarks which are searched for at the LHC.
In this case we consider components of $\tilde{R}_{2D}$ decays into $qN$ mode
with $q$ being a SM quark. Thus, the signal is two jets (heavy quarks) with
missing transverse energy that is obtained from pair production of the
leptoquarks at the LHC. Therefore, relevant constraints can be obtained from
squark searches [43]. The analysis of squark searches indicate lower limit of
the mass is around 1130 GeV for one type of squark pair production decaying
into jet and neutralino (missing energy) when the mass difference between the
squark snd neutralino is sufficiently large. We adopt the constraint for our
leptoquark mass.

The presence of DM candidates in the model sets DM relic and Direct Detection
(DD) constraints. There are only two possible DM candidates $N$ and
$\eta^{0}$. We consider $N$ as the main DM candidate. The relevant equations
for obtaining the DM relic abundance are given by

$\displaystyle\Omega_{N}h^{2}$ $\displaystyle\approx\frac{1.07\times
10^{9}\text{GeV}^{-1}}{M_{\text{pl}}}\frac{x_{F}}{\sqrt{g_{\star}}}\frac{1}{\left(a+3b/x_{F}\right)},$
(14a) $\displaystyle x_{F}$
$\displaystyle=\ln\left[c(c+2)\frac{45}{8}\frac{g_{N}}{2\pi^{3}}\frac{m_{N}M_{\text{pl}}(a+6b/x_{F})}{\sqrt{g_{\star}}\sqrt{x_{F}}}\right],$
(14b) $\displaystyle\left\langle\sigma v\right\rangle$ $\displaystyle\approx
a+b\left\langle v^{2}\right\rangle+\mathcal{O}\left(\left\langle
v^{4}\right\rangle\right)\approx a+6b/x,$ (14c)

where $x=m_{N}/T$, $F$ index stands for freeze-out temperature or $x_{F}$
value, $c$ is the order one constant determined from matching the early-time
and late-time solutions of eq. (14b). $M_{\text{pl}}=\sqrt{\hbar
c/G_{N}}=1.220\times 10^{19}$ GeV$/c^{2}$ [9] is the Planck mass. Eq. (14c) is
the non-relativistic approximation of the DM annihilation cross-section. $a$
and $b$ have the units of GeV-2 and are given explicitly for our model in Eqs.
(15). $g_{N}=2$ is the degrees of freedom (DoF) of the DM particle and
$g_{\star}$ are the relativistic DoF, in thermal equilibrium with SM sector,
at the time or temperature of DM freeze-out and $h=H_{0}/100$ km s-1 Mpc-1,
where $H_{0}=67.4\pm 0.5(73.0\pm 1.0)~{}\text{km~{}s}^{-1}~{}\text{Mpc}^{-1}$
from CMB Planck [44] (SH0ES [45]) is the Hubble parameter of the universe
today. Eq. (14a) determines the DM relic abundance for a given $a$ and $b$
coefficients, determined by the model (Eq. 15). Most up-to-date results on DM
relic abundance are those by Planck Collaboration, $\Omega_{N}h^{2}\approx
0.1200(12)$ (2018) [44]. $a$ and $b$ coefficients are generated by Feynmann
diagrams given in Fig. 3 and are explicitly given as follows

$\displaystyle a=$
$\displaystyle\frac{9m_{N}^{2}Y_{4}^{4}}{32\pi\left(m_{N}^{2}+m_{\eta}^{2}\right)^{2}}+\frac{3m_{N}^{2}Y_{2}^{4}}{8\pi\left(m_{N}^{2}+m_{\tilde{R}_{2D}}^{2}\right)^{2}},$
(15a) $\displaystyle b=$
$\displaystyle-\frac{3m_{N}^{2}Y_{4}^{4}\left(11m_{N}^{4}+30m_{N}^{2}m_{\eta}^{2}-5m_{\eta}^{4}\right)}{64\pi\left(m_{N}^{2}+m_{\eta}^{2}\right)^{4}}$
$\displaystyle-\frac{m_{N}^{2}Y_{2}^{4}\left(11m_{N}^{4}+30m_{N}^{2}m_{\tilde{R}_{2D}}^{2}-5m_{\tilde{R}_{2D}}^{4}\right)}{16\pi\left(m_{N}^{2}+m_{\tilde{R}_{2D}}^{2}\right)^{4}}.$
(15b)

(a) DM annihilation to SM leptons via $\eta$ mediated t-channel Feynmann
diagram. $\eta$ contribution to DM relic density.

(b) DM annihilation to SM quarks via $\tilde{R}_{2D}$ mediated t-channel
Feynmann diagram. $\tilde{R}_{2D}$ contribution to DM relic density.

Figure 3: Feynmann diagrams contributing to the DM annihilation cross-section.
$\eta$ and $\tilde{R}_{2D}$ contributions.

The results for the DM relic density satisfying the experimental data of [44]
are included in the plots of Figs. 4 and 5 in Sec. 6.

Next, DM DD constraint is considered. DM DD contribution in the model comes
from crossing the diagram shown in Fig. 3(b) with the $\tilde{R}_{2D}$ LQ in
the s-channel. The contribution to the DM DD cross-section from Feynmann
diagram in Fig. 3(b) is given by [46]

$\displaystyle\sigma_{\text{nucleon}}^{\text{SI}}$
$\displaystyle=\frac{\mu_{N}^{2}}{\pi}\left[\displaystyle\sum_{q=u,d,s}\left(f_{q}^{p}+f_{q}^{n}\right)\right]^{2}\frac{Y_{2}^{4}}{\left(s-m_{\tilde{R}_{2D}}^{2}\right)^{2}},$
(16a) $\displaystyle s$ $\displaystyle\simeq
m_{N}^{2}\left(1+\frac{m_{p,n}}{m_{N}}\right)^{2},$ (16b)
$\displaystyle\mu_{N}$ $\displaystyle=\frac{m_{N}m_{p,n}}{m_{N}+m_{p,n}},$
(16c)

where $p$ and $n$ correspond to proton and neutron, respectively. Meanwhile,
$f_{u,d,s}^{p,n}$ matrix elements are ([46] and references therein)

$\displaystyle f_{d}^{p}=0.033,\quad f_{u}^{p}=0.023,\quad f_{s}^{p}=0.26,$
(17a) $\displaystyle f_{d}^{n}=0.042,\quad f_{u}^{n}=0.018,\quad
f_{s}^{n}=0.26.$ (17b)

The most stringent up-to-date limit on the DM DD corresponds to the spin-
independent DM-nucleon scattering cross-section, excluding cross-sections
stronger than $9.8\times 10^{-48}$ cm2 at the DM mass of $36$ GeV/c2 presented
by the LUX-ZEPLIN data [47].

Using the constraints listed in the current section, the bounds on the model
parameters are derived and summarized in the following section.

##  6 Results

The outcomes are summarised in the two following plots. The Fig. 4 presents a
plot of DM mass, $m_{N}$, vs inert Higgs doublet mass, $m_{\eta}$, together
with incorporated DM relic (Eq. (14a)), neutrino mass, and LFV branching
ratio(Eq. (10)) constraints. The plot in Fig. 4 contains the following
information: colors of the contours correspond to the values of the Yukawa
coefficients, $Y_{4}$ from Eq. (1a), listed in the legend, solid contours
correspond to the observed DM relic density, dashed diagonal lines represent
the valid neutrino mass scale with $\lambda$ value given in the plot, finally
dash-dotted flat contours indicate the lower mass bound on the $m_{\eta}$ set
by the LFV constraint. Furthermore, plot includes the mass hierarchy
constraints $m_{N}\geq m_{\eta,\tilde{R}_{2D}}$. The Yukawa $Y_{4}$ values
were chosen of the order $\mathcal{O}(0.1)$ due to DM relic density
constraint. Lower $Y_{4}$ values $\lesssim\mathcal{O}(10^{-2})$ lead to an
overabundant DM relic density. For a given set of Yukawa values (Fig. 4), the
allowed $\lambda$(Eq. (6)) range, constrained by the LFV and DM relic, is
$\lambda\in(2.51\times 10^{-5},2\times 10^{-3})$. If $\lambda<2.51\times
10^{-5}$ LFV constraint is violated, on the other hand, if $\lambda>2\times
10^{-3}$ the DM relic cannot be satisfied. DM relic sets a lower bound on the
$Y_{2}\approx 0.576$ Yukawa value.

Figure 4: $m_{N}$ vs $m_{\eta}$ plot. Solid contours correspond to the correct
relic DM density for Yukawa $Y_{4}$ couplings listed in the plot. Dashed
colored contours represent the experimental neutrino mass value for Yukawa
$Y_{4}$ couplings listed in the plot. The dash-dotted contours signify the
lower mass bound on the $m_{\eta}$ obtained from the Br$(\mu\rightarrow
e\gamma)<4.1\times 10^{-13}$ [42] LVF constraints. Since $N$ is DM,
$m_{N}<m_{\eta,\tilde{R}_{2D}}$ limits are included as well.

The plot in Fig. 5 demonstrates the dependence of proton lifetime (Eq. (7)),
DM relic density (Eq. (14a)), DM DD (Eq. (16)), and collider constraints on
the DM mass, $m_{N}$, and LQ mass $m_{\tilde{R}_{2D}}$. The Fig. 5’s plot
comprises the following information: solid contours correspond to the observed
DM relic density with colors of the solid contours corresponding to the values
of the Yukawa coefficients, $Y_{2}$ from Eq. (1a), listed in the legend,
dashed contours represent the lower $\tilde{R}_{2D}$ LQ mass bound set by the
proton lifetime experimental limits [38, 39, 40] with colors of the dashed
contours corresponding to the values of the Yukawa coefficients, $Y_{1,3}$,
from Eq. (1a), black solid line corresponds to the CMS LHC constraint [43] on
the $\tilde{R}_{2D}$ LQ mass, finally black dashed contour indicate the lower
mass bound on the $m_{\tilde{R}_{2D}}$ set by the DM DD constraint (Eq. (16)).
Furthermore, plot includes the mass hierarchy constraints $m_{N}\geq
m_{\eta,\tilde{R}_{2D}}$. One important observation is the upper limit of
$Y_{1,3}$ Yukawas set by the proton lifetime and DM relic constraints for
$Y_{4}\approx 0.69$. Furthermore, if $Y_{4}<0.69$ then the upper limit on the
$Y_{1,3}$ Yukawas tightens down to $2.2\times 10^{-11}$.

Figure 5: $m_{N}$ vs $m_{\tilde{R}_{2D}}$ plot. Solid black contour
corresponds to the CMS LHC [43] bound on Leptoquark $\tilde{R}_{2}$. Dashed
black contour identifies the DM DD constraints obtained from LUX-ZEPLIN (LZ)
Experiment [47] 2022 Data with $Y_{2}^{u,d}=0.025$. Solid contours correspond
to the correct relic DM density for Yukawa $Y_{2}$ couplings listed in the
plot. Dashed colored contours represent the _lower bound_ on the
$m_{\tilde{R}_{2D}}$ obtained from experimental proton lifetime constraints
for $Y_{1,3}$ couplings listed in the plot. Since $N$ is DM,
$m_{N}<m_{\eta,\tilde{R}_{2D}}$ limits are included as well.

The DM DD requires the Yukawa coupling, $Y_{2}^{u,d}$, of the DM to the first
family SM quakrs to be of the order $\mathcal{O}(0.01)$ in order to satisfy
the constraint. On the other hand, the DM abundance forces Yukawas of the
order of $\mathcal{O}(0.1)$ for the DM couplings to the SM leptons and second
and third family quakrs to obtain the observed DM relic density.

The next generation proton decay search experiments, such as Hyper-Kamiokande
[48], Deep Underground Neutrino Experiment (DUNE) [49], and _paleo-detectors_
proposed in [50], will probe the proton lifetime up to the $10^{34}-10^{35}$
yr values [48, 49, 50]. Thereby, further restricting the parameter space of
the model which will play an important role on the validity of the current
model in the forthcoming future.

One of the main properties of the model is the natural smallness of the
neutrino mass and naturally long lifetime of the proton. Furthermore, the
model predicts a correlation between the smallness of the two quantities:
neutrino mass and proton decay width. This correlation is explicitly plotted
in the Fig. 6 for a given benchmark point of the model parameters. These
parameters are included in the Fig. 6.

Figure 6: $m_{N}$ vs neutrino mass and proton decay width correlation plot
with other particle masses and Yukawa couplings fixed to the values given in
the plot.

The final plot in Fig. 7 demonstrates the DM freeze-out temperature, $T_{F}$,
and $x_{F}=m_{N}/T_{F}$ parameter for a range of DM masses, $m_{N}$, and a
given benchmark point of the model parameters, which are incorporated in the
Fig. 7 as well.

Figure 7: $m_{N}$ vs freeze-out temperature ($T_{F}$) and
$x_{F}=\frac{m_{N}}{T_{F}}$ plot satisfying the experimental DM relic density
constraint with $m_{\tilde{R}_{2D}}$ LQ mass and Yukawa couplings fixed to the
values given in the plot.

##  7 Discussion

Here we discuss our findings. Firstly, constraints from neutrino mass and LFV
restricts parameters $\\{m_{N},m_{\eta},Y_{4},\lambda\\}$ as summarized in
Fig. 4. In addition interactions associated with $Y_{4}$ contribute to DM
annihilation $N\bar{N}\to\ell\bar{\ell}(\nu\bar{\nu})$ via $\eta$ exchange. We
find it is difficult to explain observed relic density only by the
interactions due to LFV constraint requiring small $Y_{4}$ or heavy $\eta$;
one can consider large hierarchy of $Y_{4}$ couplings to avoid LFV constraint
but it would not be favored for neutrino sector to fit experimental data
causing hierarchy among elements of neutrino mass matrix. We find that the
process $N\bar{N}\to q\bar{q}$ via leptoquark $\tilde{R}_{2D}$ exchange
assists in the explanation of the observed relic density. The strength of this
process is characterized by the coupling $Y_{2}$.

The parameters $\\{m_{N},m_{\eta},m_{\tilde{R}_{2D}},Y_{1,2,3}\\}$ are also
constrained by proton decay, DM direct detection, collider experiments and
relic density as summarized in Fig. 5. For $Y_{2}$ coupling, hierarchy among
components are required to satisfy collider, direct detection and proton decay
constraints, explaining relic density at the same time; we need to make
$Y_{2}$ components associated with first generation quarks to be much smaller
the other components. Also proton decay constraints require smallness of
$Y_{1,3}$ couplings.

In order to achieve the correlation between the smallness of neutrino masses
and long proton lifetime several pathways are at our hands, one of them we
discussed in the present work, others include obtaining proton decay operator
from the neutrino mass origin, and vice versa generating small neutrino masses
via the proton decay operators. These and other interesting prospects are to
be studied in the future works.

##  8 Conclusion

An enhanced model of scotogenic neutrino mass with naturally long proton
lifetime has been described. The existence of dark matter leads to a naturally
small neutrino masses and inherently long proton lifetime. Furthermore, the
smallness of neutrino masses and proton decay width has a build correlation
property. The dark matter abundance has been achieved via dark matter
annihilation to SM leptons and quarks with dark matter Yukawa couplings to SM
leptons and quarks of the order $\mathcal{O}(0.1)$, with the BSM leptoquark
mass value as low as $1200$ GeV. Dark matter direct detection forces the
Yukawa coupling of dark matter with the first family quarks to be of the order
$\mathcal{O}(0.01)$. The valid range of $2.51\times 10^{-5}<\lambda<2\times
10^{-3}$ has been obtained for the $\lambda$ coupling in the 2HDM. The
scotogenic fermionic dark matter with the mass in the range from $100$ GeV to
$10$ TeV successfully satisfies all relevant constraints, including dark
matter relic density, direct detection, neutrino mass, proton lifetime, and
lepton flavor violation. The enhanced scotogenic neutrino mass paradigm has
interconnected the three issues in particle physics: dark matter, smallness of
neutrino mass, and stability of the proton.

###### Acknowledgements.

The work was supported by the National Natural Science Fund of China Grant No.
12350410373 (O. P.) and by the Fundamental Research Funds for the Central
Universities (T. N.).

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