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126 2021 Machine Learning for Healthcare

# Medically Aware GPT-3 as a Data Generator for Medical Dialogue Summarization

Bharath Chintagunta<EMAIL_ADDRESS>Namit Katariya<EMAIL_ADDRESS>Xavier
Amatriain<EMAIL_ADDRESS>Anitha Kannan<EMAIL_ADDRESS>
###### Abstract

In medical dialogue summarization, summaries must be coherent and must capture
all the medically relevant information in the dialogue. However, learning
effective models for summarization require large amounts of labeled data which
is especially hard to obtain. We present an algorithm to create synthetic
training data with an explicit focus on capturing medically relevant
information. We utilize GPT-3 as the backbone of our algorithm and scale 210
human labeled examples to yield results comparable to using 6400 human labeled
examples ($\sim$30x) leveraging low-shot learning and an ensemble method. In
detailed experiments, we show that this approach produces high quality
training data that can further be combined with human labeled data to get
summaries that are strongly preferable to those produced by models trained on
human data alone both in terms of medical accuracy and coherency.

##  1 Introduction

With increasing usage of telehealth platforms Mann et al. (2020), large scale
ecosystems of providers and patients have become apparent. This has
exacerbated the need for comprehensive visit summaries of the medical
dialogues by the attending practitioner in order to facilitate accurate hand-
offs to other care providers or as a means of recording the interaction.
Unfortunately, having providers write summaries after each encounter is not
only time consuming but also costly, limiting the scalability of telehealth
platforms Shanafelt et al. (2016)

In these settings, an automated summarizer that can assist the practitioners
can be extremely valuable. However, an important challenge of end-to-end
medical dialogue summarization is the lack of large scale annotated datasets.
Annotation of medical dialogues is expensive and slow because they need to be
curated by trained experts. This is further compounded by the fact that
labeled data may not be publicly shared because of patient privacy concerns
and HIPAA regulations.

Recent approaches to summarization See et al. (2017); Qi et al. (2020); Zhang
et al. (2019) use transfer learning where a pre-trained model (e.g. through
self supervision of learning a language model) is fine tuned with a labeled
dataset. However, fine-tuning still requires hundreds to thousands of labeled
examples to obtain reasonable performance. Methods such as Joshi et al. (2020)
aim to partially overcome these issues through modeling strategies that
directly learn important inductive biases from smaller amounts of data. In
addition, Joshi et al. (2020) also handled data sparsity by leveraging a key
insight of sequential nature of information flow in a medical dialogue: global
summary of the dialogue can be composed from local dialogue turns (snippets).
This enables collecting training data for snippets as opposed to the full
conversation - an insight, we use in our paper as well.

Recently, OpenAI developed GPT-3, a neural language model that is capable of
natural language generation and completion of tasks like classification,
question-answering, and summarization Brown et al. (2020). The focus of that
work is to enable task-agnostic and zero-shot or low-shot performance as
opposed to a pre-trained model that needs to be fine-tuned separately on every
downstream task. In this paper, we investigate the following question: How can
a low-shot learner such as GPT-3 be leveraged to scale training data for
medical dialogue summarization models? In answering this question within the
context of GPT-3 as a black box proprietary API111https://beta.openai.com/, we
took into account multiple considerations:

  * •

Medical Correctness Joshi et al. (2020): Medical summarization warrants high
precision and therefore the summarizer should be good at capturing all the
medical information (medications, symptoms etc) discussed in the dialogue and
(2) discern all the affirmatives and negatives on medical conditions correctly
(e.g. no allergies, having a cough for 2 days).

  * •

Privacy Concerns: At inference time, an API call to external services such
GPT-3 may not always be possible due to HIPAA and privacy concerns.

  * •

Practitioner in the loop: The technique needs to be easily amenable to a
feedback loop that allows for leveraging manually curated human labels. This
feedback loop is extremely important because the diversity and the long tail
of data distribution in medical dialogue means that there can be parts of the
summary that needs to be edited by practitioners for medical correctness and
completeness. Note that these edits can be used as additional data for
improving the underlying model.

Taking into account these considerations, this paper makes the following
contributions (Figure 1 for a quick overview):

  * •

We introduce a medically-aware GPT-3 data labeler, GPT-3-ENS , that combines
medical knowledge and an ensemble of GPT-3 for the purpose of medical dialogue
summarization.

  * •

We introduce the idea of using GPT-3-ENS as a dataset generator to facilitate
learning an in-house summarization model. Our experiments show that we can
obtain the same performance as that of human labeled dataset with 30x smaller
amount of human labeled data. With only 210 expert curated summaries and GPT-3
as a labeled data simulator, we can mimic the performance of a summarization
model trained on 6400 expert curated summaries.

  * •

By combining generated datasets from GPT-3-ENS with a human labeled dataset,
we show that we can obtain better performance than models trained on either
one of the data sources.

The rest of the paper is structured as follows: § 3 discusses related work, §
5 introduces our approach, § 6 and § 7 describe our datasets and metrics
respectively while § 8 illustrates our experiments. We end the paper with § 9
discussing our conclusions and future work.

Figure 1: Overview of our proposed approach: we train models on a mix of
GPT-3-ENS synthesized and human labeled data to get performance better than
models trained on either of the sources

##  2 Generalizable Insights about Machine Learning in the Context of
Healthcare

One of the main challenges in using deep learning for healthcare is the lack
of large annotated datasets. It is usually costly and time-consuming to
collect a large labeled dataset because annotations need to be provided by
trained healthcare professionals. As deep models usually require a large
amount of data to perform accurately and robustly, this deters their
widespread application in healthcare. So, it is essential to develop low-shot
models in healthcare i.e. models that can do well given a small number of
labeled examples. In parallel, there has been a lot of progress in development
of large scale models leveraging web-scale data, such as GPT-3, that show good
low-shot performance. However, these models can be noisy, particularly in the
medical domain, so we need approaches that mitigate this noise but are still
able to leverage these models’ strengths. In this context, our approach of
infusing medical knowledge in pretrained models such as GPT-3 to generate
high-quality synthetic labels is an idea with wide applicability in low-
resource settings like healthcare.

If pretrained models can be used to generate accurate labels, can they be
directly leveraged for the task at hand? In many settings, they probably can
but particularly in healthcare, this is nuanced. ML models in healthcare can
learn and improve over time only if they are amenable to feedback loops i.e.
they can be retrained with labels that are corrected / edited by medical
practitioners. Moreover, if the model making the predictions is owned by a
third party, privacy protocols (e.g. HIPAA) mandate that either they obey the
same privacy protocols or that data be deidentified before being sent to such
external services. Both these necessitate the need for a different approach.
We introduce one such approach where we infuse medical knowledge into an
external non-HIPAA compliant model (GPT-3) and leverage it as a data generator
to obtain a large training set, to then train an in-house model. Since the
data exposed to GPT-3 is fixed and small (in our experiments, GPT-3 only saw
210 examples), it can be ensured to be privacy protected. Our proposed
approach to develop an in-house model has two advantages (1) It can be used at
inference time without the practical constraint of data de-identification and
(2) It lends itself well to the aforementioned practitioner-in-the-loop
setting.

##  3 Related work

Summarization Emergence of sequence to sequence models and attention
mechanisms Sutskever et al. (2014) has led to rapid progress on extractive
Nallapati et al. (2017) , abstractive Nallapati et al. (2016); Zhang et al.
(2019) and hybrid models See et al. (2017); Gu et al. (2016) for
summarization. Much of the recent work has shown these models to generate
near-human coherent summaries while retaining reasonable factual correctness.

Dialogue summarization: While most neural summarization has focused on news
corpora, recent work has tried to tackle unique challenges associated with
summarizing dialogues. Goo and Chen (2018) proposes using dialogue history
encoders based on the type of dialogue section to inform the generation. Liu
et al. (2019a) propose using key points as a means of categorizing sections of
dialogue.

Medical dialogue summarization Existing work Alsentzer and Kim (2018); Zhang
et al. (2018); Liu et al. (2019b); Krishna et al. (2020a, b); Joshi et al.
(2020) in this space focuses on effective summarization by incorporating
medical knowledge from a modeling perspective. Our work also focuses on
incorporating medical knowledge from a data labeling perspective. We show how
we leverage pretrained language models and low-shot learning Brown et al.
(2020) to collect labeled data for medical dialogue summarization. We also
show how this data can improve performance over models that are trained solely
on existing human labeled data.

##  4 Background: Can GPT-3 serve as a medical summarizer?

Ignoring the privacy concerns and practitioner-in-the-loop considerations, we
first explore whether GPT-3 Brown et al. (2020) is a good medical summarizer
by itself.

GPT-3 takes as input a priming context to perform the task on a previously
unseen example. Priming context refers to the text description of a task and a
few demonstrations of the task being accomplished (in our case, that would be
dialogue snippet summarization).

Table 1 column 2 provides examples of summaries generated by the GPT-3 model.
We can clearly see that it misses a number of important pieces of information
in the snippets – first, missing medical concepts making the summary unusable
(Rows 1-2). Second, the model may not always get the affirmations correct (Row
3). Third, the summary may repeat redundant information from the doctor’s
queries (Row 4).

Based on these observations, we might prematurely conclude that GPT-3 can not
be used for medical summarization task. However, our key observation in
exploring GPT-3 is that it is sensitive to the priming context (also reported
in Liu et al. (2021)), as the model does not learn but just adheres to the
examples given. As we show in 5, we exploit this variability in GPT-3 output
via ensembling and infusion of medical knowledge so that it can be used as a
part of an effective low-shot learning approach to medical summarization.

Snippet |  GPT-3 |  GPT-3-ENS
---|---|---
DR: Thank you so much for sharing. Are you on any regular medications for that? PT: My only regular medication is birth control -Apri. Low dosage. |  On birth control. |  Only regular medication is birth control -Apri. Low dosage.
DR: You had mentioned the headache starting after stopping the medications. Have you had similar headache while stopping medications before? PT: Yes PT: Well that’s a complicated question PT: I’m not really sure DR: Okay, no worries. Wanted to know a bit more since you had mentioned about them. That’s all. |  Yes. Stopping medications before. |  Yes, has headache while stopping medications before. Is not sure.
DR: Do you have pain when you notice penile discharge? PT: no i’m not |  Did not notice penile discharge. No pain. |  Doesn’t have pain when noticing penile discharge.
DR: I have a few questions to ask. How long have you been having this vaginal discomfort? PT: only a few days, since like friday or saturday maybe |  I have a few questions to ask. How long has she been having vaginal discomfort? |  Has been having vaginal discomfort for only a few days, since friday or saturday.
Table 1: Input dialogue snippets along with summaries generated by GPT-3 in
column 2 and our approach, GPT-3-ENS , in column 3.

##  5 Infusing Medical Knowledge in GPT-3 for use as a Data Generator

We are interested in a model that uses only a small amount of human labeled
data to learn an effective medical dialogue summarizer. At the same time, we
want such a model to be used in a practical practitioner-in-the-loop setting
where medical correctness and patient privacy are of paramount importance.

In order to achieve these goals, we propose a two-pronged approach

  1. 1.

Introduce GPT-3-ENS  where we infuse medical knowledge into GPT-3 and use it
within an inner loop to make it effective at medical summarization.

  2. 2.

Leverage GPT-3-ENS as a data generator to obtain a large training set
222Unlike data at inference time, training data is fixed and can be ensured to
be privacy protected to train an in-house medical dialogue summarization
model. Such an in-house model can be used at inference time without the
practical constraints related to protecting patient privacy that would require
full de-identification to be applied in any conversation, if we were to access
the GPT-3 service. It also lends itself well to the practitioner-in-the-loop
setting.

###  5.1 GPT-3-ENS : Medically-aware ensemble of GPT-3

As discussed in 4, GPT-3 is quite sensitive to the priming context. While one
approach may be to provide GPT-3 with the most informative context for a task,
this itself is a daunting task and can potentially be tackled if we had a
large number of labeled examples (which is the exact problem we want to tackle
with GPT-3).

Drawing on the learning from vast literature in ensembling techniques c.f.
Bishop et al. (1995), our first key insight is that if we can generate
multiple summaries from GPT-3 using a variety of priming contexts, then we
should be able to ensemble these outputs to identify the summary that is ideal
for the dialogue. This insight leads to a question on how to ensemble multiple
text summaries. The answer to this question relies on the core requirement for
medical summarization: we care about the coverage of medical concepts
mentioned and therefore the best ensembling function is the one that returns
the summary with the most medical information in the dialog input.

In Algorithm 1 we provide our approach to the medically aware GPT-3 ensemble
GPT-3-ENS . We assume access to a small set of labeled examples $\mathcal{L}$.
For each input dialog snippet, $T$, we get $K$ summaries, by invoking GPT-3
each time with $N$ examples sampled randomly without replacement from
$\mathcal{L}$. We also assume access to a medical entity extractor that can
discern the medical concepts from both the dialogue snippet and the summary.
The algorithm returns the best summary that has the highest recall in terms of
capturing the medical concepts in the dialogue. For this purpose, we use an
in-house medical concept extractor MedicalEntityRecognizer that can identify
medical concepts from a given piece of text. This extractor has access to the
universe of medical concepts based on Unified Medical Knowledge Systems
333https://www.nlm.nih.gov/research/umls/index.html, which includes patient
symptoms, disorders, laboratory tests and medications. Note that any medical
entity recognizer (cf. Fu et al. (2019) and references therein) that has
coverage for all these types of medical concepts found in medical
conversations can be used.

Algorithm 1 Medically aware GPT-3 ensemble summarizer (GPT-3-ENS )

1:dialogue snippet $T$, ensembling trials $K$, universe $\mathcal{L}$ of
labeled examples, medical entity extractor MedicalEntityRecognizer, GPT3

2:$C^{*}\leftarrow\texttt{MedicalEntityRecognizer}(T)$

3:for $i\leftarrow 1,\cdots,K$ do

4: $S\leftarrow\text{sample $N$ examples from $\mathcal{L}$}$

5: $\texttt{summary}_{i}\leftarrow\texttt{GPT3}(S,T)$

6: $C_{i}\leftarrow\texttt{MedicalEntityRecognizer}(\texttt{summary}_{i})$

7:$\texttt{summary}_{\texttt{best}}\leftarrow\texttt{summary}_{\operatorname*{arg\,max}_{i}\frac{|C_{i}\cap
C^{*}|}{|C^{*}|}}$

8:return $\texttt{summary}_{best}$

Reconsider Table 1 for qualitative comparison between GPT-3 and GPT-3-ENS . We
can see that summaries obtained using GPT-3-ENS capture the medical concepts
comprehensively (shown in bold) and also have better grammatical structure. We
also quantitatively validate the summaries on a small data set distinct from
what is used for priming(see § 7.2 for guidelines). In Figure 2, based on
doctor evaluation, we can see that GPT-3-ENS is significantly better at
summarization than GPT-3 .

Figure 2: Doctor evaluation of which among GPT-3 and GPT-3-ENS summaries they
considered “best” showing that GPT-3-ENS is a better approach for labeling

###  5.2 GPT-3-ENS as a data labeler

We use GPT-3-ENS described in 5.1 as our labeled data generator. In
particular, we use our approach to collect a large amount of labeled examples
that serve as inputs to training an off-the-shelf summarization model. This
resolves the concern of using GPT-3 in a real world application where the
patient’s conversation (in its raw form) needs to be exchanged with an
external third party such as OpenAI/GPT-3 which may not have design/privacy
regulations around HIPAA. In our approach, however, with the help of experts,
it is easy to ensure that the dialogues that will used for priming as well as
in the training set are chosen following privacy protocols.

##  6 Datasets

We collected a random subset of medical conversation dialogues from our chat-
based telemedicine platform. Often medical conversation follows a linear
ordering of medical history gathering (understanding patient symptoms) that
enables creating the summary of the dialog by stitching together summaries of
the snippets in chronological order Joshi et al. (2020). Therefore, we split
each dialogue into a series of local dialogue snippets using a simple
heuristic: the turns between two subsequent questions by a physician
corresponds to a snippet. The length of these snippets ranged anywhere from
two turns (a physician question and patient response) to ten turns.

We had medical doctors444These are the same doctors who practice on the same
telemedicine platform. summarize these snippets. The doctors were asked to
summarize the sections as they would for a typical clinical note by including
all of the relevant history taking information. If a local snippet did not
contain any history taking information it was excluded from annotations. For
example in the beginning or end of conversations there may be turns that are
purely greetings and not part of the patient history taking process. Further
some snippets maybe purely educational in nature and are excluded as well. We
eventually obtained a total of 6900 labeled snippet-summary pairs.

Human labeled dataset train/test split: From the 6900 labeled snippet-summary
pairs (denoted as $H_{6900}$), we generated a randomly sampled test set
$T=500$ that we use in all our evaluations.

The dataset $H_{6900}-T$ is used to generate the priming dataset for GPT-3
related models as well as the datasets we use to train our summarization
models.

GPT-3-ENS dataset: Let GCF${}_{p}^{k}$ be the dataset of size $p$ generated
using GPT-3-ENS with $k$ ensembling trials. To generate dataset GCFK=k, we
require $\\{\text{H}_{n}\\}_{i=1}^{k}$ datasets (note the independence on
$p$), and thus $n\times k$ labeled examples for priming. These $n\times k$
examples are randomly sampled from the universe of human labeled examples
$H_{6900}-T$. In our experiments, we sample without replacement so that no
examples are reused across the $k$ tries. To allow comparison between our
experiments with different $K$ values, we use the same seed for random
sampling.

##  7 Evaluation Metrics

Multiple studies have shown that automated metrics in NLP do not always
correlate well to human judgments as they may not fully capture coherent
sentence structure and semantics Roller et al. (2020); Kryściński et al.
(2019). Since medical dialogue summarization would be used to assist health
care, it is important for doctors to evaluate the quality of the output.

###  7.1 Automated metrics

While we measure model performance on standard metrics of ROUGE Lin (2004)
555We use the following package with default configuration:
https://github.com/google-research/google-research/tree/master/rouge, we also
measure a model’s effectiveness in capturing the medical concepts that are of
importance, and their negations Joshi et al. (2020)

Medical Concept Coverage: The concept coverage set of metrics captures the
coverage of medical terms in the model’s output summary with respect to the
ground truth. In particular, let $\mathcal{C}$ be the set of medical concepts
in the reference summary and $\hat{\mathcal{C}}$ be the set of concepts in the
summary output by the model. Then, $\textrm{Concept
recall}=\frac{\sum_{n=1}^{N}|\hat{\mathcal{C}}^{(n)}\cap\mathcal{C}^{(n)}|}{\sum_{n=1}^{N}|\mathcal{C}^{(n)}|}$
and $\textrm{Concept
precision}=\frac{\sum_{n=1}^{N}|\hat{\mathcal{C}}^{(n)}\cap\mathcal{C}^{(n)}|}{\sum_{n=1}^{N}|\hat{\mathcal{C}}^{(n)}|}$.

We use these to compute a Concept F1666Note if there are no concepts detected
in the snippet and summary by the entity extractor, then a conservative F1
score of 0 is given for that example. We use an in-house medical entity
extractor to extract medical concepts in the summary. Medical concepts in the
decoded summary that weren’t present in the original conversation would be
false positives and vice versa for false negatives.

Negation Correctness: To measure the effectiveness of the model to identify
the negated status of medical concepts, we use Negex Harkema et al. (2009) to
determine negated concepts. Of the concepts present in the decoded summary, we
evaluate precision and recall on whether the decoded negations were accurate
for the decoded concepts and compute a negation F1${}^{\ref{footnote f1}}$.

###  7.2 Doctor Evaluation

We also had doctors, who serve patients on our telehealth platform, evaluate
the summaries produced by the models. Given the local dialogue snippets and
the generated summary, we asked them to evaluate the extent to which the
summary captured factually correct and medically relevant information from the
snippet. Depending on what percentage of the concepts were correctly mentioned
in the decoded summary of the provided snippet, the doctors graded the
summaries with _All_ (100%), _Most_ (at least 75%), _Some_ (at least 1 fact
but less than 75%), _None_ (0%) labels.

We also formulated a comparison task where given summaries generated by
different models and the associated dialogue, they were asked which summary
was the "best" from a usability perspective. Usability was defined as whether
the summary could stand in as a replacement for reading the dialogue snippet
i.e. whether it captures the correct concepts from the snippet and whether the
negations are accurate. The doctors had the ability to use “all” and “none” in
this task depending on if all models being compared captured a good summary or
if none of them did.

To avoid bias, the doctors do not know the model that produced the summary in
both the experiments. In the comparison task, the summaries were provided in
randomized order so that there is no bias in the order of presentation of the
summaries.

##  8 Experiments and Results

Additional models considered: To evaluate the efficacy of GPT-3-ENS  as a
source of labeled data generator, we considered models with distinct objective
functions for abstractive and hybrid (abstractive/extractive) summarization.
We used PEGASUS  Zhang et al. (2019) for abstractive summarization and Dr.
Summarize which we denote as DRSUM  Joshi et al. (2020) for extractive
summarization. For DRSUM , we also use their best performing variant (referred
as 2M-PGEN in Joshi et al. (2020)) which penalizes generator loss and favors
extractive copying.

Implementation Details: We used GPT-3 via the API released by
OpenAI777https://beta.openai.com/. Maximum response length was set to 128
tokens, temperature to 0.6 and presence and frequency penalties both set to 0.
For GPT-3-ENS , we use $K=10$ ensembling trials for all our experiments,
unless otherwise specified. We observed that $N=21$ was the maximum number of
examples we could prime GPT-3 with given the maximum context window length of
2048 tokens for the API. We therefore fix the size of our priming dataset to
be 21 in all experiments which invoke GPT-3. Hence we set $L$ to be a random
subset of 210 examples from $H_{6900}-T$.

We followed parameter settings for DRSUM from Joshi et al. (2020) for
pretraining on the CNN-Dailymail dataset. We then fine-tuned on our
summarization task dataset with a batch size of 16, source_max_tokens = 400,
response_max_tokens = 200 and max_grad_norm clipped at 2.0, for two epochs
with a learning rate of 0.15 using Adagrad optimizer.

We used the PEGASUS implementation that is pretrained on CNN-
Dailymail888https://huggingface.co/google/pegasus-cnn_dailymail provided by
Wolf et al. (2020). We fine-tuned it on our summarization task dataset with an
effective batch size of 256, source_max_tokens = 512, response_max_tokens =
128 for two epochs using
Adafactor999https://huggingface.co/transformers/main_classes/optimizer_schedules.html#adafactor-
pytorch optimizer at the default settings in Hugging Face. For both PEGASUS
and DRSUM , we used a beam size of four for decoding.

###  8.1 Training summarization models using data labeled by GPT-3-ENS

We compare PEGASUS and DRSUM trained on human labeled data H6400 and GPT-3-ENS
synthesized data GCF${}^{K=10}_{6400}$. Note that synthesizing
GCF${}^{K=10}_{6400}$ needed all of $21\cdot 10=210$ human labeled examples,
where $21$, as a reminder, is the maximum number of inputs that can be used
for priming.

Table 2 compares quantitative performance of PEGASUS and DRSUM trained on
these two datasets. The main observation is that with only 210 human labeled
examples, our approach GPT-3-ENS is able to generate a large amount of
training data for both pre-trained summarization models, PEGASUS and DRSUM ,
in such a manner that they yield comparable (or better perfomance) than if
they had been trained with only 6400$(\sim$30x) human labeled examples.

For PEGASUS , the summarization performance improves drastically compared to
model fine-tuned using only the human labeled data. We hypothesize that data
generated from GPT-3-ENS can serve as quality training data for abstractive
models such as PEGASUS but not so much for hybrid models such as DRSUM due to
GPT-3 being a generative language model. The summaries written by our human
doctors have writing structure similar to that of a hybrid summarization model
such as DRSUM that is more extractive in nature. This can explain why DRSUM
did not show performance gain when using generated data from GPT-3-ENS . The
key, however, is that it still did perform on par.

Models Train Data Source Metrics Negation F1 Concept F1 ROUGE-L F1 PEGASUS
$H_{6400}$ 21.09 35.96 55.59 $GCF_{6400}^{k=10}$ 28.89 40.02 53.43
$GCF_{12800}^{k=10}$ 26.70 40.21 56.66 $GCF_{25600}^{k=10}$ 28.61 40.58 58.44
DRSUM $H_{6400}$ 26.75 39.95 52.70 GCF${}_{6400}^{k=10}$ 24.29 37.55 48.47
GCF${}_{12800}^{k=10}$ 26.66 38.49 49.18 GCF${}_{25600}^{k=10}$ 26.08 39.47
50.85

Table 2: Automated evaluation of summarization models trained with different
data labeling methodologies. Note that the amount of human labeled data is
still pretty low (210), compared to 6400 when we do not use our approach.

In the same Table 2, we also present the results with increased amounts of
data (12800 and 25600) from GPT-3-ENS . There is little or no further
improvement in the automated metrics of concept and negation F1. However,
ROUGE-L F1 improves reflecting the improvements in coherency of the summaries.
We leave this area as future work to explore.

###  8.2 Effect of combining human labeled data with data labeled by GPT-3-ENS

Since GPT-3 relies on limited local priming context ($N=21$) it may not be
agile in providing robust summaries for a multitude of variations in snippets,
focusing on the exploitation part of the exploration-exploitation trade-off.
We hypothesize that best summaries then will be synthesized by a model trained
on a dataset with human and GPT-3-ENS labeled examples. To evaluate this, we
introduced a mixing parameter $\alpha$, the ratio of GPT-3-ENS labeled
examples to human labeled examples. For instance, with 6400 human labeled
examples, $\alpha=0.5$ implies the dataset contains 6400 human labeled
examples along with $0.5*6400=3200$ GPT-3-ENS generated examples. We
experiment with $\alpha=0.5,1,2,3$.

From Table 4, we observe that for both PEGASUS and DRSUM , mixture of human
labeled and GPT-3-ENS data consistently improves almost all automated metrics
for all $\alpha$ values101010Note here that the claim is not that increasing
$\alpha$ improves metrics but that mixing GPT-3-ENS and human labeled data
improves metrics over models trained only using human data. We leave it as a
future work on how to trade-off between human and GPT-3-ENS labeled data. The
lift in metrics is lower for DRSUM , again illustrating the idea we
highlighted in § 8.1 of GPT-3-ENS data being more amenable to abstractive
models such as PEGASUS than for hybrid or extractive-biased models such as
DRSUM . Table 3 provides qualitative comparison between summaries generated by
each of these models.

For simplicity, we chose the smallest GPT-3-ENS mix i.e. $\alpha=0.5$ for
human evaluation where we ask doctors to evaluate summaries from model trained
on human, GPT-3-ENS and human+GPT-3-ENS data. Figure 3 and Figure 4 show that
doctors prefer summaries from the model trained on the mixture data over those
produced by models trained on human or GPT-3-ENS data alone, in terms of
amount of medical information captured as well as the overall quality of the
summary. Furthermore, Figure 3(b) also shows that for PEGASUS , doctors prefer
the summaries from a model trained on GCF${}^{K=10}_{6400}$ (which needed only
210 human labeled examples) over those produced by a model trained on 6400
human labeled examples.

Snippet |  Model trained on H6400 |  Model trained on GCF${}^{K=10}_{6400}$ |  Model trained on H6400+GCF${}^{K=10}_{3200}$
---|---|---|---
DR: Have you ever been tested for any underlying health conditions such as diabetes, hypothyroidism or polycystic ovarian syndrome? PT: No PT: I have been told I have prediabetes |  Has not been tested for any underlying health conditions. |  Hasn’t tested for any underlying health conditions such as diabetes, hypothyroidism or polycystic ovarian syndrome |  Has not been tested for any underlying health conditions. Has been told has prediabetes.
DR: Do you have pus appearing discharge from the site? PT: Yes. If the bubbles pop it leaks out a watery substance |  Has pus appearing from the site. |  Pus appearing from the site |  Pus discharge from the site. If bubbles pop it leaks out a substance.
Table 3: Input conversation snippets along with summaries generated by models
trained on different data

Models Train Data Source Metrics Negation F1 Concept F1 ROUGE-L F1 PEGASUS
$H_{6400}$ 21.09 35.96 55.59 $\alpha=0.5$ $H_{6400}+GCF_{3200}^{K=10}$ 30.14
43.49 62.45 $\alpha=1$ $H_{6400}+GCF_{6400}^{K=10}$ 30.70 43.73 60.63
$\alpha=2$ $H_{6400}+GCF_{12800}^{K=10}$ 29.43 41.02 59.85 $\alpha=3$
$H_{6400}+GCF_{25600}^{K=10}$ 31.93 44.68 61.05 DRSUM $H_{6400}$ 26.75 39.95
52.70 $\alpha=0.5$ $H_{6400}+GCF_{3200}^{K=10}$ 27.51 40.46 53.39 $\alpha=1$
$H_{6400}+GCF_{6400}^{K=10}$ 27.18 40.36 51.00 $\alpha=2$
$H_{6400}+GCF_{12800}^{K=10}$ 27.19 40.68 53.07 $\alpha=3$
$H_{6400}+GCF_{25600}^{K=10}$ 26.33 39.89 52.29

Table 4: Combining human labeled datasets with datasets generated using our
proposed approach Figure 3: Doctor evaluation of amount of medical information
covered by summaries provided by PEGASUS models and which ones they considered
“best” Figure 4: Doctor evaluation of amount of medical information covered by
summaries provided by DRSUM models and which ones they considered “best”

##  9 Conclusion

We introduced a medically-aware GPT-3 data labeler, GPT-3-ENS , for the task
of medical conversation summarization. At the heart of the approach is a
medically aware ensembling criterion that ensembles multiple summaries for an
input from a powerful low-shot learner such as GPT-3. We showed that this
approach can generate quality training data for medical dialogue summarization
models while ensuring medical correctness. We show that using a very small
number of human labeled examples, 210, we are able to produce more medically
correct and better quality summaries than using roughly thirty times as many
human labeled examples for two different summarization models. In this work we
used a simple ensembling technique that dialogue summaries should retain all
the medical information discussed in the dialogue.

Limitations and Future Work In the future, we would like to improve our
ensembling function to take into account other medical priors such as
affirmations and importance/relevance of the information in the dialog or even
come up with a different way of ensembling altogether. Another active area is
to understand and model trade-off between the amount of human and GPT-3-ENS
labeled data in the training dataset. Thirdly, we can investigate ways to
prime GPT-3 in an unsupervised manner that leads to better coverage of latent
space of the domain. Lastly, we would also like to investigate other
applications in healthcare where the idea of using large pretrained models to
generate synthetic labels can be useful.

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##  Appendix A GPT-3 Prompt

We utilize a fairly simple prompt to have GPT-3 generate summaries. Each
example (snippet_text, summary_text) is concatenated to the empty string with
the following transformation:"{snippet_text}[SUMMARIZED]{summary_text}[STOP]"
to form the prompt. We seperate the conversational turns in snippet_text with
the "[SEP]" token. Table 5 shows a prompt that would be generated and used to
prime GPT-3 given two examples. As mentioned in § 8 in our experiments we use
21 examples to generate a prompt.

Snippet | Summary | Prompt
---|---|---
|  PT: Today spit out a bit of
---
mucus and noticed a bit of
blood.
DR: Okay, how long have
you been on these medications?
PT: About 2 years
| Has been on these medications
---
for about 2 years.
| Today spit out a bit of mucus and noticed a
---
bit of blood.[STOP] Okay, how long have you
been on these medications?[SEP] About 2 years
[SUMMARIZED] Has been on these medications
for about 2 years.[STOP]
|  DR: Is the bleeding
---
from the anal opening and
not the vagina? Has something
similar happened before?
PT: yes from the anal opening
| The bleeding is from the
---
anal opening.
| Is the bleeding from the anal opening and not the
---
vagina? Has something similar happened before?
[SEP]yes from the anal opening[SUMMARIZED]
The bleeding is from the anal opening.[STOP]

Table 5: Prompt for GPT-3 given two examples


# Minimal Vectorlike Model in Supersymmetric Unification

Sibo Zheng1,2<EMAIL_ADDRESS>1Department of Physics, Chongqing
University, Chongqing 401331, China
2Department of Physics, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Compared to the minimal supersymmetric standard model, an extension by
vectorlike fermions is able to explain the Higgs mass while retains the grand
unification. We investigate the minimal vectorlike model by focusing on the
vectorlike electroweak sector. We firstly derive the mass spectrum in the
electroweak sector, then calculate the one-loop effects on the Higgs physics,
and finally explore either vectorlike or neutralino dark matter. Collider
constraints are briefly discussed.

##  I Introduction

With the discovery of Higgs scalar Higgsmass1 ; Higgsmass2  at the Large
Hadron Collider (LHC), identifying the nature of this scalar, which reveals
the origin of electroweak symmetry breaking (EWSB), is one of prior tasks at
this facility. There are various scenarios in the literature that naturally
explain the EWSB, among which supersymmetry (SUSY) has attracted much
attention since the discovery. Nevertheless, the prospective on the minimal
supersymmetric standard model (MSSM) is rather pessimistic. Some reasons for
this include that the mass parameters related to the third-generation squarks
are probably at least of order several TeVs MSSMHiggs1 ; MSSMHiggs2 , and also
signals of neutralino dark matter aren’t yet observed at any dark matter
direct detection facilities MSSMDark1 ; MSSMDark2 .

The status of MSSM motivates a diversity of extensions, of which vectorlike
(VL) fermions are interesting because of a few features as follows.

  * •

First, the VL fermions are one of viable extensions which retain the
perturbative unification 1711.05362 ; 1706.01071 , similar to a well-known
example - the next-to-minimal supersymmetric model (NMSSM).

  * •

Second, the issue of naturalness imposed by the Higgs mass in the MSSM can be
resolved by new radiative correction to the Higgs mass due to the VL fermions
of order TeV scale 0910.2732 ; MO ; Babu . It differs from the NMSSM, as in
the later one the correction occurs at the tree level.

  * •

Last, the stringent constraint on the neutralino dark matter from dark matter
direct detection will be modified due to be presence of neutral fermions in
the VL sector.

This paper is devoted to study the minimal VL model as described in Table.1.
We will explore this model by focusing on the VL leptons therein, the matter
content of which is composed of one singlet and two doublets. A key point in
this model is that the VL leptons directly couple to the Higgs doublets rather
than mix with the SM leptons, which suggests that they may play important
roles in both the Higgs physics and dark matter.

VL model |  Representation
---|---
Minimal | $\mathbf{1}+\mathbf{5}+\mathbf{\bar{5}}$
LND 0910.2732  | $\mathbf{1}+\mathbf{1}+\mathbf{5}+\mathbf{\bar{5}}$
VL 4th-gen Babu  |  $\mathbf{1}+\mathbf{5}+\mathbf{\bar{5}}+\mathbf{10}+\mathbf{\bar{10}}$
Table 1: TeV-scale VL models and their representations under
${\mathrm{S}U(5)}$.

The paper is organized as follows. In Sec.II, we introduce the minimal VL
model, derive the EWSB conditions, and calculate mass spectrum in the
electroweak sector for our purpose. Sec.III is devoted to estimate effects on
the Higgs physics, where the one-loop radiative correction to Higgs mass and
the modification on Higgs self coupling are presented in Sec.IIIA and
Sec.IIIB, respectively. Sec.IV is devoted to discuss the dark matter physics
in the presence of VL leptons, where in Sec.IVA we will show the VL dark
matter is excluded by the large Yukawa coupling as required by the Higgs mass,
whereas in Sec.IVB the parameter space of neutralino dark matter is expanded
due to the relax of Higgs mass constraint. In Sec.V, we briefly discuss the
collider constraints on the VL leptons. Finally, we conclude in Sec.VI.

##  II The Model

In the minimal VL model, the matter content contains a singlet $N$, two down-
type triplets $D$ and $\bar{D}$, and two VL doublets $L$ and $\bar{L}$ with
hyper charge $1/2$ and $-1/2$, respectively. They constitute a
$\mathbf{5}_{V}$ and a $\bar{\mathbf{5}}_{V}$ representation of SU(5)
consistent with grand unification 1706.01071 ; 1711.05362 .

The superpotential for the VL electroweak sector reads as,

$\displaystyle{}W_{VL}=kH_{u}\bar{L}N-hLH_{d}N+M_{L}\bar{L}L+\frac{1}{2}M_{N}N^{2}$
(1)

where $M_{L,N}$ refer to VL masses and $k$ and $h$ are Yukawa coupling
constants. The superpotential for MSSM is given by,

$\displaystyle{}W_{\text{MSSM}}=Y_{u}q\bar{u}H_{u}+Y_{d}q\bar{d}H_{d}+Y_{e}l\bar{e}H_{d}+\mu
H_{u}H_{d}.$ (2)

Although the matter content in Eq.(1) is similar to that of NMSSM with two
additional doublets Fayet , they are different as what follows. Following our
purpose as mentioned above, we assume that the conservative $R$-parities of
$\mathbf{5}_{V}$ and $\bar{\mathbf{5}}_{V}$ that contain $L$ and $\bar{L}$ are
the same as those of the $\mathbf{5}_{H}$ and $\bar{\mathbf{5}}_{H}$ that
include the Higgs doublets $H_{u}$ and $H_{d}$. Moreover, in order to ensure
that the low-energy effective superpotential is precisely described by Eq.(1)
and Eq.(2), we impose a $Z_{2}$ parity, under which $N$, $\mathbf{5}_{V}$ and
$\bar{\mathbf{5}}_{V}$ referring to the VL sector are odd but
$\bar{\mathbf{5}}_{M}$, $\mathbf{10}_{M}$, $\mathbf{5}_{H}$ and
$\bar{\mathbf{5}}_{H}$ referring to the MSSM sector are even. The $Z_{2}$
parity is spontaneously broken by the vacuum expectation values (vevs) in the
VL sector. Since the triplets $D$ and $\bar{D}$ affect the phenomenological
analysis111 Neither they directly couple to the Higgs doublets nor mix with
the neutralinos/VL electroweak fermions. Thus, both the corrections to the
Higgs physics and dark matter analysis are expected to be small. Note,
although perturbative unification favors VL triplet mass $M_{D}\sim M_{L}$, it
may be still intact even for a moderate mass splitting between $M_{D}$ and
$M_{L}$. Because it is not only sensitive to the mass spectrum of VL sector
but also that of MSSM sector, and even intermediate physics between the weak
scale and unification scale. only in terms of their interaction with heavy
triplet fields in $\mathbf{5}_{H}$ and $\bar{\mathbf{5}}_{H}$, we will simply
neglect them.

Apart from the superpotential, the soft mass Lagrangian
$\mathcal{L}_{\text{soft}}$ is also expanded as,

$\displaystyle{}-\mathcal{L}_{\text{soft}}$ $\displaystyle\supset$
$\displaystyle-\mathcal{L}^{\text{MSSM}}_{\text{soft}}+m^{2}_{L}\mid
L\mid^{2}+m^{2}_{\bar{L}}\mid\bar{L}\mid^{2}+m^{2}_{N}\mid N\mid^{2}$ (3)
$\displaystyle+$
$\displaystyle\left(kA_{k}H_{u}\bar{L}N-hA_{h}LH_{d}N+\text{H.c}\right),$

where $m_{L,N,D}$ and $A_{k,h}$ represent soft mass parameters, and
$\mathcal{L}^{\text{MSSM}}_{\text{soft}}$ contains MSSM soft masses such as

$\displaystyle{}-\mathcal{L}^{\text{MSSM}}_{\text{soft}}=m^{2}_{H_{u}}\mid
H_{u}\mid^{2}+m^{2}_{H_{d}}\mid H_{d}\mid^{2}+(bH_{u}H_{d}+\text{H.c}).$

Following the notation and conventions in 9709356 , we express the Higgs and
VL doublets as, respectively,

$\displaystyle{}H_{u}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}H^{+}_{u}\\\ H^{0}_{u}\\\
\end{array}\right)=\left(\begin{array}[]{c}H^{+}_{u}\\\
\upsilon_{u}+\frac{1}{\sqrt{2}}(H^{0}_{uR}+iH^{0}_{uI})\\\ \end{array}\right)$
(8) $\displaystyle H_{d}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}H^{0}_{d}\\\ H_{d}^{-}\\\
\end{array}\right)=\left(\begin{array}[]{c}\upsilon_{d}+\frac{1}{\sqrt{2}}(H^{0}_{dR}+iH^{0}_{dI})\\\
H^{-}_{d}\\\ \end{array}\right)$ (13)

and

$\displaystyle{}L$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}E^{+}\\\ \eta\\\
\end{array}\right)=\left(\begin{array}[]{c}E^{+}\\\
v_{L}+\frac{1}{\sqrt{2}}(\eta_{R}+i\eta_{I})\\\ \end{array}\right)$ (18)
$\displaystyle\bar{L}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\bar{\eta}\\\ E^{-}\\\
\end{array}\right)=\left(\begin{array}[]{c}\bar{v}_{L}+\frac{1}{\sqrt{2}}(\bar{\eta}_{R}+i\bar{\eta}_{I})\\\
E^{-}\\\ \end{array}\right)$ (23)

Similarly, we write singlet $N$ as

$\displaystyle{}N=n+\frac{1}{\sqrt{2}}(N_{R}+iN_{I}).$ (24)

Mass parameters $\upsilon_{u,d}$, $v_{L}$, $\bar{v}_{L}$ and $n$ denote vevs.

These vevs are determined by the minimization of scalar potential $V$ as given
by

$\displaystyle{}V=V_{F}+V_{D}-\mathcal{L}_{\text{soft}}$ (25)

where

$\displaystyle{}V_{F}$ $\displaystyle=$ $\displaystyle\mid\mu
H_{d}+k\bar{L}N\mid^{2}+\mid\mu H_{u}-hLN\mid^{2}$ $\displaystyle+$
$\displaystyle\mid M_{L}\bar{L}-hH_{d}N\mid^{2}+\mid
kH_{u}N+M_{L}L\mid^{2}+\mid kH_{u}\bar{L}-hLH_{d}+M_{N}N\mid^{2},$
$\displaystyle V_{D}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{1}+g^{2}_{2}}{8}(\mid H_{u}\mid^{2}-\mid
H_{d}\mid^{2}+\mid L\mid^{2}-\mid\bar{L}\mid^{2})^{2}+\frac{g^{2}_{2}}{2}\mid
H_{u}H^{{\dagger}}_{d}+L\bar{L}^{{\dagger}}\mid^{2}.$ (26)

Here, $g_{1}$ and $g_{2}$ refers to $U(1)_{Y}$ and $SU(2)_{L}$ gauge coupling
constant, respectively. Substituting Eq.(18) and Eq.(24) into Eq.(25) gives
rise to the EWSB conditions:

$\displaystyle{}0$ $\displaystyle=$ $\displaystyle
F_{N}(-k\bar{v}_{L})+F_{\bar{L}}(kn)+F_{H_{d}}\mu-
kA_{k}n\bar{v}_{L}+m^{2}_{H_{u}}\upsilon_{u}-\frac{1}{2}M^{2}_{Z}\cos
2\beta\upsilon_{u}(1-\delta)-b\cot\beta\upsilon_{u}$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle
F_{N}(hv_{L})+F_{L}(-hn)+F_{H_{u}}\mu+hA_{h}nv_{L}+m^{2}_{H_{d}}\upsilon_{d}+\frac{1}{2}M^{2}_{Z}\cos
2\beta\upsilon_{d}(1-\delta)-b\tan\beta\upsilon_{d}$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle
h^{2}n^{2}v_{L}+F_{N}(h\upsilon_{d})+hA_{h}n\upsilon_{d}+m^{2}_{L}v_{L}-\frac{1}{2}M^{2}_{Z}\cos
2\gamma v_{L}\delta$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle
k^{2}n^{2}\bar{v}_{L}+F_{N}(-k\upsilon_{u})-kA_{k}n\upsilon_{u}+m^{2}_{\bar{L}}\bar{v}_{L}+\frac{1}{2}M^{2}_{Z}\cos
2\gamma\bar{v}_{L}\delta$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle
F_{N}M_{N}+F_{L}(-h\upsilon_{d})+F_{\bar{L}}(k\upsilon_{u})+n(k^{2}\bar{v}^{2}_{L}+h^{2}v^{2}_{L})-kA_{k}\upsilon_{u}\bar{v}_{L}+hA_{h}\upsilon_{d}v_{L}+m^{2}_{N}n,$
(27)

where $F_{i}$ refer to the $F$-terms related to chiral superfield
$i=\\{N,L,\bar{L},H_{u,d}\\}$. For simplicity, we have neglected small cross
terms from $D$-terms. In Eq.(II), $\tan\beta$, $\tan\gamma$ and $\delta$ are
defined as

$\displaystyle{}\tan\gamma$ $\displaystyle=$
$\displaystyle\frac{v_{L}}{\bar{v}_{L}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta=\frac{v^{2}_{L}+\bar{v}^{2}_{L}}{(\upsilon/\sqrt{2})^{2}},$
$\displaystyle\tan\beta$ $\displaystyle=$
$\displaystyle\frac{\upsilon_{u}}{\upsilon_{d}},~{}~{}~{}~{}1-\delta=\frac{\upsilon^{2}_{u}+\upsilon^{2}_{d}}{(\upsilon/\sqrt{2})^{2}},$
(28)

with the expectation value for the weak scale $\upsilon\simeq 246$ GeV.

With the conditions of EWSB described in Eq.(II), one can directly derive the
scalar and fermion mass spectrum in the electroweak sector. We refer the
reader to appendix A and B for scalar mass spectrum such as CP-even, CP-odd,
and CP-charged scalars, and fermion mass spectrum such as neutralinos and
charginos, respectively. In the next section, we will discuss constraints on
the model parameters from precision measurements on the Higgs couplings.

##  III Higgs Physics

###  III.1 Higgs Couplings

The precision measurement on the Higgs couplings is one of major tasks at the
LHC, for it reveals the pattern of EWSB in the sense that different new
physics models predict different sets of Higgs couplings. The LHC has verified
the SM-like Higgs coupling to SM fermions such as $b$ 1709.07497  and $\tau$
1501.04943  as well as couplings to SM vector bosons such as $W$ 1412.2641 .
These measurements tell us to what extension 1303.3879 ; 1606.02266 ;
1809.10733  $\upsilon$ is deviated from its SM reference value. Such data will
be further improved in higher level of precision at the future HL-LHC
1310.8361  or ILC 1310.0763  in preparation.

|  Parameter |  Fit
---|---|---
$k_{t}$ | $\frac{1}{\sqrt{1-\delta}}\frac{\mid\mathcal{O}_{11}\mid}{s_{\beta}}$ | $0.81^{+0.19}_{-0.15}$
$k_{b}$ | $\frac{1}{\sqrt{1-\delta}}\frac{\mid\mathcal{O}_{12}\mid}{c_{\beta}}$ | $0.74^{+0.33}_{-0.29}$
$k_{\tau}$ | $\frac{1}{\sqrt{1-\delta}}\frac{\mid\mathcal{O}_{21}\mid}{c_{\beta}}$ | $0.84^{+0.19}_{-0.18}$
$k_{W}$ | $\mid\sqrt{1-\delta}(s_{\beta}\mathcal{O}_{11}+c_{\beta}\mathcal{O}_{12})+\sqrt{\delta}(s_{\gamma}\mathcal{O}_{13}+c_{\gamma}\mathcal{O}_{14})\mid$ | $0.95^{+0.14}_{-0.13}$
$k_{Z}$ | $\mid\sqrt{1-\delta}(s_{\beta}\mathcal{O}_{11}+c_{\beta}\mathcal{O}_{12})+\sqrt{\delta}(s_{\gamma}\mathcal{O}_{13}+c_{\gamma}\mathcal{O}_{14})\mid$ | $1.05^{+0.16}_{-0.16}$
vev | $\upsilon\sqrt{1-\delta}$ | $231^{+13}_{-15}$
Table 2: LHC constraints from global fits to SM-like Higgs couplings 1412.8662
and weak scale $\upsilon$ 1809.10733  at $68\%$ CL. Mass parameter is in unit
of GeV.

The Higgs couplings to SM particles in our model are given as,

$\displaystyle{}y_{t}$ $\displaystyle=$
$\displaystyle\frac{m_{t}}{\upsilon}\frac{\mathcal{O}_{11}}{\sqrt{1-\delta}s_{\beta}},$
$\displaystyle y_{b}$ $\displaystyle=$
$\displaystyle\frac{m_{b}}{\upsilon}\frac{\mathcal{O}_{12}}{\sqrt{1-\delta}c_{\beta}},$
$\displaystyle y_{\tau}$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}}{\upsilon}\frac{\mathcal{O}_{12}}{\sqrt{1-\delta}c_{\beta}},$
$\displaystyle y_{W}$ $\displaystyle=$
$\displaystyle\frac{2M^{2}_{W}}{\upsilon}\left[\sqrt{1-\delta}(s_{\beta}\mathcal{O}_{11}+c_{\beta}\mathcal{O}_{12})+\sqrt{\delta}(s_{\gamma}\mathcal{O}_{13}+c_{\gamma}\mathcal{O}_{14})\right],$
$\displaystyle y_{Z}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{Z}}{\upsilon}\left[\sqrt{1-\delta}(s_{\beta}\mathcal{O}_{11}+c_{\beta}\mathcal{O}_{12})+\sqrt{\delta}(s_{\gamma}\mathcal{O}_{13}+c_{\gamma}\mathcal{O}_{14})\right],$
(29)

where the constant coefficients are the SM values 222The Higgs couplings
$y_{W}$ and $y_{Z}$ receive contributions both from the expansions of the
kinetic energy terms of Higgs doublets $H_{u,d}$ and VL doublets $L$ and
$\bar{L}$, which are proportional to the vevs $\upsilon_{u,d}$ and
$v_{L},\bar{v}_{L}$, respectively. Following the definitions in Eq.(II), we
read the forms of $y_{W}$ and $y_{Z}$.. $\mathcal{O}_{ij}$ are components of
the orthogonal matrix $\mathcal{O}$ which diagonalizes the CP-even mass matrix
$\mathcal{M}^{2}_{S}$ in appendix A. In terms of $\mathcal{O}_{ij}$ the SM-
like Higgs scalar $h_{1}$ can be written as,

$\displaystyle{}h_{1}=\mathcal{O}_{11}H^{0}_{uR}+\mathcal{O}_{12}H^{0}_{dR}+\mathcal{O}_{13}\eta_{R}+\mathcal{O}_{14}\bar{\eta}_{R}+\mathcal{O}_{15}N_{R}.$
(30)

Unlike in the case of two Higgs doublets, where it is convenient to
parameterize the couplings of SM-like Higgs scalar in term of two free angles
$\beta$ and $\alpha$, in our model there are five new parameters $m_{N}$,
$m_{L,\bar{L}}$ and $A_{k,h}$ and three new EWSB conditions as shown in
Eq.(II). It implies that in our case there are four free parameters related to
the Higgs couplings, which are subject to the constraints from the global fits
to the LHC measurements on the Higgs couplings as summarized in Table.2. In
the following analysis, we choose the set of parameters $\tan\beta$,
$\mathcal{O}_{11}$, $\mathcal{O}_{12}$ and $\delta$ for later discussions.

The best fit value for $\delta$ has changed from the earlier $244$ GeV in
ref.1303.3879  to $233$ GeV in ref.1606.02266  and $231$ GeV in ref.1809.10733
. The latest constraint on $\delta$ in Table.2 suggests the following
parameter range,

$\displaystyle{}0\leq\delta\leq 0.22,$ (31)

which implies that the EWSB is dominated by the Higgs doublet vevs. So, it is
rational to ignore the small $\sqrt{\delta}$ term in the scaling factors
$k_{W}$ and $k_{Z}$ in Table.2. With these approximations, we obtain the
constraint on $\mathcal{O}_{11}$ and $\mathcal{O}_{12}$ in Table.2

$\displaystyle{}0.78\leq\mathcal{O}^{2}_{11}+\mathcal{O}^{2}_{12}\leq 1.0,$
(32)

after we perform a numerical scan within the region $10\leq\tan\beta\leq 50$.

###  III.2 Higgs Self Coupling

Apart from the precision measurements on the Higgs couplings above, a probe of
Higgs self interaction is also useful in identifying the nature of EWSB. In
the SM, the tri-Higgs and quartic Higgs coupling reads as, respectively,

$\displaystyle{}\lambda^{\text{SM}}_{3h}$ $\displaystyle=$
$\displaystyle\frac{3m^{2}_{h}}{\upsilon},$
$\displaystyle\lambda^{\text{SM}}_{4h}$ $\displaystyle=$
$\displaystyle\frac{3m^{2}_{h}}{\upsilon^{2}},$ (33)

where the self coupling $\lambda_{3h}$ is a key parameter to determine the SM
Higgs pair production at LHC 9603205 ; GB ; 9805244 ; 1305.7340 ; 1309.6594 ,
which mainly arises from gluon gluon fusion. The cross section for Higgs pair
production is altered in new physics such as MSSM and NMSSM even at tree level
9603205 ; 9805244 ; 1206.5816 ; 1304.3670 ; 1305.6397 ; 1504.06932 .

Figure 1: The contour of $\lambda_{3h}/\lambda^{\text{SM}}_{3h}$ as function
of $\delta$ and $\mathcal{O}_{11}$ for $\tan\beta=20$ and
$\mathcal{O}_{12}=0.05$. The shaded region satisfies all constraints in the
Table.2, where triangle region on the right side of green curve is in the
reach of HL-LHC ($14$ TeV, 3ab-1) 1310.8361 .

In our case, both $\lambda_{3h}$ and $\lambda_{4h}$ receive contributions from
$V_{F}$ and $V_{D}$, which are given by,

$\displaystyle{}\lambda_{3h_{1}}$ $\displaystyle=$
$\displaystyle\frac{3M^{2}_{Z}}{\upsilon}\left[\sqrt{1-\delta}(\mathcal{O}^{2}_{11}-\mathcal{O}^{2}_{12})(s_{\beta}\mathcal{O}_{11}-c_{\beta}\mathcal{O}_{12})+\sqrt{\delta}(\mathcal{O}^{2}_{13}-\mathcal{O}^{2}_{14})(s_{\gamma}\mathcal{O}_{13}-c_{\gamma}\mathcal{O}_{14})\right],$
$\displaystyle\lambda_{4h_{1}}$ $\displaystyle=$
$\displaystyle\frac{3M^{2}_{Z}}{\upsilon^{2}}\left[(\mathcal{O}^{2}_{11}-\mathcal{O}^{2}_{12})^{2}+(\mathcal{O}^{2}_{13}-\mathcal{O}^{2}_{14})^{2}\right].$
(34)

From Eq.(32), one observes that the components $\mathcal{O}_{11}$ and
$\mathcal{O}_{12}$ constitute most of SM-like Higgs within region with
moderate or large $\tan\beta$. In contrast, the other components
$\mathcal{O}_{1j}$ ($j=3-5$) can be ignored. In Eq.(III.2), we have neglected
contributions to $\lambda_{3h}$ and $\lambda_{4h}$ due to $V_{F}$ which are at
least of order $\mathcal{O}^{2}_{1j}$. As expected, the results in Eq.(III.2)
reduce to those of MSSM under the limit $\delta\rightarrow 0$,

$\displaystyle{}\lambda^{\text{MSSM}}_{3h_{1}}$ $\displaystyle=$
$\displaystyle\frac{3M^{2}_{Z}}{\upsilon}\cos(2\alpha)\sin(\alpha+\beta),$
$\displaystyle\lambda^{\text{MSSM}}_{4h_{1}}$ $\displaystyle=$
$\displaystyle\frac{3M^{2}_{Z}}{\upsilon^{2}}\cos^{2}(2\alpha).$ (35)

The probe of Higgs self interaction is thus useful in discriminating this
model from the others.

Following the constraints on $\mathcal{O}_{11}$ and $\mathcal{O}_{12}$ in the
Table.2, we show in Fig.1 the contour of
$\lambda_{3h}/\lambda^{\text{SM}}_{3h}$ for $\tan\beta=20$ and
$\mathcal{O}_{12}=0.05$, with the shaded region satisfying all constraints in
the Table.2. Note the ratio $\lambda_{3h}/\lambda^{\text{SM}}_{3h}$ is more
sensitive to scaling factor $\delta$ and $\mathcal{O}_{11}$ rather than
$\mathcal{O}_{12}$ for large $\tan\beta$. Under the limit $\delta\rightarrow
0$, the ratio approaches to the maximal value
$\lambda_{3h}^{\text{max}}/\lambda^{\text{SM}}_{3h}$ $\sim 0.52$. With
$\delta\neq 0$, a suppression on the ratio appears. We find that the triangle
region on the right side of green curve is in the reach of HL-LHC ($14$ TeV,
3ab-1) 1310.8361 , whereas the left side will be still invisible in the
foreseeable future.

###  III.3 Higgs Mass

The soft mass parameters in Eq.(3) lead to mass splittings between the fermion
and its scalar partner masses in VL superfield $L$, $\bar{L}$ and $N$. It can
be extracted via evaluating the one-loop correction to effective potential CW
,

$\displaystyle{}\Delta V=2\sum^{3}_{i=1}[F(M^{2}_{b_{i}})-F(M^{2}_{f_{i}})],$
(36)

with $F(x)=x^{2}[\ln(x/Q^{2})-3/2]/64\pi^{2}$. For simplicity, we consider the
universal soft masses $m_{L}=m_{\bar{L}}=m_{N}=m$, universal VL masses
$M_{L}=M_{\bar{L}}=M_{N}=M$ and large $\tan\beta$ limit. Meanwhile, we take
the good approximation $v_{L}=\bar{v}_{L}=0$. The neutral scalar mass squared
matrix, which determines the mass eigenvalues $M^{2}_{b_{i}}$ in Eq.(36), is
given by (see Eq.(56))

$\displaystyle{}M^{2}_{b}\simeq\left(\begin{array}[]{ccc}M^{2}+m^{2}&0&kM\upsilon_{u}\\\
&M^{2}+m^{2}+k^{2}\upsilon^{2}_{u}&-kX_{k}\upsilon_{u}\\\
&*&M^{2}+m^{2}+k^{2}\upsilon^{2}_{u}\\\ \end{array}\right)~{}$ (40)

where $X_{k}\simeq M+A_{k}-\mu\cot\beta$. The neutral fermion mass squared
matrix, which gives the mass eigenvalues $M^{2}_{f_{i}}$ in Eq.(36), reads as
(see also Eq.(72))

$\displaystyle{}M^{2}_{f}\simeq\left(\begin{array}[]{ccc}M^{2}+k^{2}\upsilon^{2}_{u}&0&kM\upsilon_{u}\\\
&M^{2}&kM\upsilon_{u}\\\ &*&M^{2}+k^{2}\upsilon^{2}_{u}\\\
\end{array}\right),$ (44)

Substituting the mass eigenvalues in Eq.(40) and Eq.(44) into Eq.(36), one
obtains the one-loop correction to SM-like Higgs mass

$\displaystyle{}\Delta
M^{2}_{h}\simeq\frac{1}{4\pi^{2}}k^{4}\upsilon^{2}\sin^{4}\beta\left[\ln(x)+\frac{1}{8}\left(\frac{\lambda(y)}{x}-\lambda_{*}\right)-\frac{5}{192}\left(\frac{\lambda^{2}(y)}{x^{2}}-\lambda^{2}_{*}\right)\right]$
(45)

where $x=1+m^{2}/M^{2}$, $y=X^{2}_{k}/M^{2}$ and $\lambda_{*}=\lambda(1)$,
with $\lambda(y)$ defined as

$\displaystyle{}\lambda(y)=4+2y+2\sqrt{y^{2}+4y}.$ (46)

As clearly seen in Eq.(45), $\Delta M^{2}_{h}\rightarrow 0$ under the
degenerate mass limit $x\rightarrow 1$ and $y\rightarrow 1$, where the scalar
and fermion masses are degenerate. Conversely, large $\Delta M^{2}_{h}$ is
expected in the situation with large $x$ and certain value of $y$.

Figure 2: The contour of Higgs mass with $\Delta
M_{h}=\sqrt{(125~{}\text{GeV})^{2}-M^{2}_{Z}}$ as function of Yukawa coupling
constant $k$ and ratio $x=1+m^{2}/M^{2}$ that describes the mass splitting
between scalars and fermions in the VL electroweak sector. The red and black
curves refer to the minimal and non-minimal VL model, respectively.

Shown in Fig.2 is the contour of Higgs mass with correction $\Delta
M_{h}=\sqrt{(125~{}\text{GeV})^{2}-M^{2}_{Z}}$. The red curves represent the
minimal VL model, which are compared with the non-minimal VL model 0910.2732
as shown in black curves. In individual case, the dashed and solid one refers
to the small- and large-mixing effect, respectively. There are a few comments
in order. $i)$, $X_{k}/M$, which is fixed by the condition of maximal-mixing
effect, is equal to $12x/5$ and $2(3x-1)$ in the minimal and non-minimal VL
model, respectively. $ii)$, The discrepancies between the two models are small
(large) in the situation with maximal (small)-mixing effect. $iii)$, All of
four patterns in Fig.2 favor large value of $x$. This suggests a large mass
splitting ($m/M\sim 10$) between the scalars and fermions in the VL
electroweak sector, if the Yukawa coupling $k$ is required to be in the
perturbative region in high energy scales. For VL fermions of order $\sim 1$
TeV, the VL scalars have mass of order $\sim 10$ TeV, which implies that the
later ones have no role to play either at the LHC or dark matter experiments.

##  IV Dark Matter

In our model, the number of neutral fermions is seven, with four from the MSSM
neutralino sector (for review, see, e.g.9506380 ) and the other three from the
VL sector. The mass matrix for them is shown in Eq.(72), where the complexity
can be relatively reduced by taking the decoupling limit $v_{L}=\bar{v}_{L}=0$
as favored by precise measurements on the Higgs couplings. Moreover, a small
but nonzero $n$ is sufficient to yield desired decays in the VL electroweak
sector, with the first-order approximation to which the mass matrix in Eq.(72)
is further divided into two separate parts - the neutralino mass matrix $A$
and the VL mass matrix $B$.

###  IV.1 VL Dark Matter

For VL dark matter $\tilde{\chi}^{0}_{1}$ is decomposed as,

$\displaystyle{}\tilde{\chi}^{0}_{1}=N^{*}_{15}\tilde{\bar{\eta}}+N^{*}_{16}\tilde{\eta}+N^{*}_{17}\tilde{N},$
(47)

where for simplicity we have neglected the neutralinos in Eq.(72). The mass
matrix $M_{\chi}$ is reduced to matrix $B$. Since all of scalars in the VL
sector are decoupled, the Lagrangian in Eq.(1) most relevant for this
situation is given by333We will write both charged and neutral fermions in
4-component notation, e.g.,
$\chi^{+}_{E}=(\tilde{E}^{+},\overline{\tilde{E}^{-}})$ and
$\tilde{\chi}^{0}_{1}=(\chi^{0}_{1},\overline{\chi^{0}_{1}})$.,

$\displaystyle{}\mathcal{L}_{\text{VL}}$ $\displaystyle\supset$ $\displaystyle
k(H^{+}_{u}\tilde{E}^{-}-H^{0}_{u}\tilde{\bar{\eta}})\tilde{N}-h(H^{0}_{d}\tilde{\eta}-H^{-}_{d}\tilde{E}^{+})\tilde{N}$
(48) $\displaystyle-$
$\displaystyle\frac{g}{4c_{W}}\overline{\tilde{\eta}}\gamma^{\mu}\gamma_{5}\tilde{\eta}Z_{\mu}+\frac{g}{4c_{W}}\overline{\tilde{\bar{\eta}}}\gamma^{\mu}\gamma_{5}\tilde{\bar{\eta}}Z_{\mu}$
$\displaystyle-$
$\displaystyle\frac{g}{\sqrt{2}}\overline{\tilde{E}^{+}}\gamma^{\mu}P_{L}\tilde{\eta}W^{+}_{\mu}+\frac{g}{\sqrt{2}}\overline{\tilde{E}^{-}}\gamma^{\mu}P_{R}\tilde{\bar{\eta}}W^{-}_{\mu}.$

From Eq.(48), the VL $\tilde{\chi}^{0}_{1}$ annihilation is mediated by $Z$
boson, SM Higgs $h_{\text{SM}}$ and the charged fermion $\tilde{E}^{\pm}$,
with SM final states $\bar{f}f$, $ZZ$, $WW$ and $h_{\text{SM}}h_{\text{SM}}$.

Figure 3: Constraints on the parameter space of VL dark matter with $k=1.2$
and $\tan\beta=20$. We have shown the relic density of dark matter (red), the
latest Xenon1T SI limit Xenon1T  (green dotted), LUX SD constraint LUXSD
(green solid), $Z$ invisible decay limit Zdecay  (blue solid), and Higgs
invisible decay limit hdecay  (blue dotted) simultaneously. The contours of VL
dark matter masses in unit of GeV are shown in gray dotted curves. Regions on
the top of green dotted curve, on the left side of green solid curve, and
below the blue curves are excluded.

Eq.(48) also indicates that this dark matter model is a specific example of
singlet-doublet dark matter 0510064 ; 0705.4493 ; 0706.0918 ; 1109.2604 ;
1311.5896 ; 1505.03867 . The key feature in our model is that the coupling
$k\simeq 1.1-1.2$ due to the observed Higgs mass. The other coupling $h$ can
be neglected for $\tan\beta\geq 10$, in which case we are left with two mass
parameters $M_{L}$ and $M_{N}$. Using these mass parameters, one obtains the
dark matter couplings to Higgs and $Z$ 1505.03867  given as respectively,

$\displaystyle{}c_{h}$ $\displaystyle\simeq$
$\displaystyle-\frac{k^{2}}{2}\frac{m_{\tilde{\chi}^{0}_{1}}\upsilon}{M^{2}_{L}+\frac{k^{2}}{2}\upsilon^{2}+2M_{N}m_{\tilde{\chi}^{0}_{1}}-3m^{2}_{\tilde{\chi}^{0}_{1}}},$
$\displaystyle c_{Z}$ $\displaystyle\simeq$
$\displaystyle-\frac{k^{2}}{2}\frac{M_{Z}\upsilon(m^{2}_{\tilde{\chi}^{0}_{1}}-M^{2}_{L})}{(m^{2}_{\tilde{\chi}^{0}_{1}}-M^{2}_{L})^{2}+\frac{k^{2}}{2}\upsilon^{2}(m^{2}_{\tilde{\chi}^{0}_{1}}+M_{L}^{2})}.$
(49)

Shown in Fig.3 are the constraints on the parameter space projected to the
two-parameter plane of $M_{L}-M_{S}$ with $k=1.2$. The red curve therein
refers to the relic density of VL dark matter $\Omega h^{2}\simeq 0.12$. We
show the latest limits Xenon1T ; LUXSD  on the spin-independent (SI) and spin-
dependent (SD) dark matter-p/n scattering cross sections in green dotted and
green solid plots, respectively, where we have used the results
$\sigma_{SI}\simeq c^{2}_{h}\times(2.11\times 10^{3})$ zb and
$\sigma^{n}_{SD}\simeq c^{2}_{z}\times(8.97\times 10^{8})$ zb. In order to be
complete, indirect constraints are also shown in Fig.3. Since the VL leptons
do not directly mix with the SM leptons, constraints such as lepton flavor
violations are weaker than the electroweak precision measurements 0803.4008
on the $Z$ boson and Higgs decay widths. In Fig.3, the experimental bounds
$\Delta\Gamma_{Z}\leq 2$ MeV Zdecay  and $\Delta\Gamma_{h}\leq 0.16\Gamma_{h}$
hdecay  are shown in blue solid and blue dotted curves, respectively,
parameter regions below which are excluded.

The combination of these limits exclude parameter regions on the top of green
dotted curve and on the left side of green solid curve. It turns out that
while VL dark matter with mass larger than $200$ GeV may be still consistent
with the LUX SD limit, it is totally excluded by the latest Xenon1T limit. The
main reason for it is that large $k\sim 1.2$ and large $\tan\beta$ lead to too
large $c_{h}$ in Eq.(IV.1) to evade the Xenon1T limit 444Of course, this
statement is not true in the situation where smaller $k$ can fit the observed
Higgs mass when the one-loop correction to the Higgs mass is dominated by the
stop parameters in the MSSM sector rather than the the electroweak fermions in
the VL sector.. In the above analysis, we have handled the dark matter relic
density by an analytic treatment similar to ref.1505.03867  instead of
numerical calcualtions such as MicrOMEGAs 1407.6129 . In particular, we have
used our previous results in ref.1711.10097 .

###  IV.2 Neutralino Dark Matter

Although the VL dark matter is excluded, it still has a role to play in the
case of neutralino dark matter. A neutralino dark matter
$\tilde{\chi}^{0}_{i}$ mainly arises from following components:

$\displaystyle{}\tilde{\chi}^{0}_{i}=N^{*}_{i1}\tilde{B}^{0}+N^{*}_{i2}\tilde{W}^{0}+N^{*}_{i3}\tilde{H}^{0}_{d}+N^{*}_{i4}\tilde{H}^{0}_{u},$
(50)

whose relic density is affected by the VL sector in the following ways.

Firstly, $\tilde{\chi}^{0}_{1}$ pair may annihilate into the VL final states
such as $\tilde{E^{+}}\tilde{E^{-}}$, through either the $s$-channel exchange
of $Z$ boson/Higgs scalar $h_{i}$ or the $t$\- and $u$-channel exchange of VL
scalars $E^{\pm}$. However, this channel is not kinetically allowed, since the
masses of $\tilde{E^{\pm}}$ are larger than $m_{\tilde{\chi}^{0}_{1}}$ when
$\tilde{E^{\pm}}$ share the same $R$-parity with the neutralino
$\tilde{\chi}^{0}_{1}$. Moreover, the $t$ and $u$-channel exchange of VL
scalars $E^{\pm}$ are both suppressed by large VL scalar masses555If there are
new sources to uplift the Higgs mass, the VL scalar masses can be of order
sub-TeV, in which situation such process deserves a detailed study 1608.00283
; 1810.07224 ..

Second, the VL sector can mediate new Feynman diagrams for neutralino dark
amtter annihilation:

  * •

Fermions $\tilde{E}^{\pm}$ mediate $t$\- or $u$\- channel Feynman diagrams for
neutralino annihilation into $W^{\pm}$, with the neutralino-$\chi^{\pm}_{E}$-W
vertex given by

$\displaystyle{}\mathcal{L}\supset-\frac{g}{\sqrt{2}}W^{-}_{\mu}(N_{61}P_{L}-N_{51}P_{R})\overline{\tilde{\chi}^{0}_{1}}\gamma^{\mu}\chi^{+}_{E}+\text{H.c}.$
(51)

  * •

Fermions $\tilde{\eta}$ and $\tilde{\bar{\eta}}$ mediate $t$\- or $u$\-
channel Feynman diagrams for neutralino annihilation into $Z$ bosons, with the
neutralino-$\tilde{\eta}/\tilde{\bar{\eta}}$-Z vertex read as

$\displaystyle{}\mathcal{L}\supset-\frac{g}{4c_{W}}Z_{\mu}\overline{\tilde{\chi}^{0}_{1}}\gamma^{\mu}\gamma_{5}\left(N_{61}\tilde{\eta}-N_{51}\tilde{\bar{\eta}}\right)$
(52)

  * •

Fermions $\tilde{\eta}$, $\tilde{\bar{\eta}}$ and $\tilde{N}$ mediate $t$\- or
$u$-channel Feynman diagrams for neutralino annihilation into $hh$, with the
neutralino-$\tilde{\eta}/\tilde{\bar{\eta}}/\tilde{N}$-$h_{\text{SM}}$ vertex
in the MSSM modified by

$\displaystyle{}\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{1}{\sqrt{2}}(-kN_{51}s_{\beta}+hN_{61}c_{\beta})\overline{\tilde{\chi}^{0}_{1}}\tilde{N}h_{\text{SM}}+\frac{1}{\sqrt{2}}N_{71}(-ks_{\beta}\tilde{\bar{\eta}}+hc_{\beta}\tilde{\eta})\overline{\tilde{\chi}^{0}_{1}}h_{\text{SM}}$
(53)

Eq.(51) to Eq.(53) determine the deviation from ordinary neutralino, which
rely on mixings between $\tilde{\chi}^{1}_{0}$ and VL sector as described by
components $N_{i1}$ with $i=5-7$. With the mass parameter $m\sim$ several
TeVs, the conditions of EWSB in Eq.(II) imply that $\langle n\rangle$ is at
most of order $\sim 10$ GeV and $\delta$ is less than $\sim 0.01$. Thus,
$N_{i1}$ reaches to its maximal values in the case of higgsino-like
$\tilde{\chi}^{1}_{0}$.

Figure 4: In the case of VL doublet-dominated correction, the ratio
$\delta\sigma_{\text{ann}}/\sigma_{\text{ann}}$ as function of higgsino-like
neutralino mass $m_{\tilde{\chi}^{0}_{1}}$ for $-h=k=1.2$, $\mid
M_{L}-m_{\tilde{\chi}^{0}_{1}}\mid<M_{Z}$, and different values of
$n=\\{5,15,25,30\\}$ GeV, where $\sigma_{\text{ann}}\simeq 3\times
10^{-27}\text{cm}^{3}/s$.

We divide the effects on higgsino-like $\tilde{\chi}^{1}_{0}$ into the VL
doublets- and singlet-dominance. In the case of VL doublet-dominated
correction, $M_{L}$ is smaller than $M_{N}$. The effect on the relic density
of higgsino-like $\tilde{\chi}^{0}_{1}$ mainly arises from the
neutralino-$\tilde{\eta}/\tilde{\bar{\eta}}$-Z vertex, which also contributes
to the SD cross section. The first-order solution to the reduced $4\times 4$
mass matrix in Eq.(72) only composed of Higgs doublets and VL doublets is
given by

$\displaystyle{}N^{\tilde{H}_{d(u)}}_{61}$ $\displaystyle\simeq$
$\displaystyle-kn/(\mu+M_{L}),$ $\displaystyle N^{\tilde{H}_{d(u)}}_{51}$
$\displaystyle\simeq$ $\displaystyle-(+)kn/(\mu+M_{L}),$ $\displaystyle
N^{\tilde{H}_{d(u)}}_{71}$ $\displaystyle\simeq$ $\displaystyle 0,$ (54)

for $h=-k$ and $M_{N}>M_{L}$.

Substituting Eq.(IV.2) into Eq.(52), we show in Fig.4 the maximal correction
to dark matter annihilation cross section $\sigma_{\text{ann}}$ that occurs in
the mass region $\mid M_{L}-m_{\tilde{\chi}^{0}_{1}}\mid<M_{Z}$ for $k=1.2$
and different values of $n=\\{5,15,25,30\\}$ GeV. In the mass range of
$m_{\tilde{\chi}^{0}_{1}}=250-1000$ GeV, the ratio
$\delta\sigma/\sigma_{\text{ann}}$ varies from $6\%$ ($n=30$ GeV) to less than
$0.1\%$ ($n=5$ GeV). It is sensitive to $n$ since $\delta\sigma$ is
proportional to factor $(n/\mu+M_{L})^{4}$. In the case of singlet-dominated
correction, similar order of $\delta\sigma/\sigma_{\text{ann}}$ is expected.

The modifications on the neutralino relic density, the SD and the SI cross
sections, which are in percent level, suggest that the parameter space of
neutralino dark matter is mainly expanded due to relaxing the Higgs mass
constraint e.g. by lowering the values of $\tan\beta$ or gaugino masses in the
case of universal soft masses.

##  V Collider Detection

Collider detections on VL electroweak fermions depend on their interactions
with SM particles. If mixed with SM leptons such as $e$, $\mu$ or $\tau$, VL
leptons can be constrained by mixing-relevant processes at collider
constraints 1506.01291 ; 0107015 ; 1510.03456 . These constraints depend on
both the VL lepton mass and its mixing with the SM leptons. In our case, the
VL electroweak fermions $\chi_{E}^{\pm}$ ($\tilde{\eta}$,
$\tilde{\bar{\eta}}$) directly mix with charginos (neutralinos) rather than SM
leptons. The strategies of searching them at colliders thus differ from the
case of lepton mixings.

Signals | Background
---|---
$\chi_{E}^{+}\chi_{E}^{-}$ 1403.5294  | jets + leptons +$E^{\mathrm{m}iss}_{T}$
$\tilde{\eta}$ or $\overline{\tilde{\eta}}$ pair 1409.3168  | jets+ (leptons) +$E^{\mathrm{m}iss}_{T}$
$\chi_{E}^{\pm}$+$\tilde{\eta}$ or $\chi_{E}^{\pm}$+$\overline{\tilde{\eta}}$ 1501.07110 ; 1803.02762  | (jets)+ leptons +$E^{\mathrm{m}iss}_{T}$
Table 3: The SM backgrounds for productions of VL electroweak fermion pairs at
the LHC, where $E^{\mathrm{m}iss}_{T}$ refers to missing transverse momentum.

The VL electroweak fermions can be produced via chargino (neutralino) decay
when their masses are beneath chargino (neutralino) masses. Alternatively,
they just imitate the electroweak productions of charginos (neutralinos). The
main difference between these two processes are that the cross section for the
former case depends on both the VL lepton masses and their mixings with
chargino or neutralinos, while the cross section for the later one mainly
relies on the VL electroweak fermion masses. Here, we stick to the later case,
where the VL electroweak fermion pairs decay and are produced in terms of SM
$Z$-, $W$\- and Higgs-boson. Under the basis of gauge eigenstates as shown in
Eq.(48), the interaction vertexes of these processes are fixed by the SM gauge
interactions, so the numbers of events for VL lepton pairs are mainly
sensitive to their mass parameters $M_{L}$ and $M_{N}$. We show in Table.3 the
backgrounds for the productions of various VL electroweak fermion paris at the
LHC, where mass bounds on neutralinos and charginos can be found.
$m_{\tilde{\chi}^{\pm}_{1}}$ or $m_{\tilde{\chi}^{0}_{2}}$ up to $\sim 550$
GeV is excluded for $m_{\tilde{\chi}^{0}_{1}}$ less than $\sim 100$ GeV, and
they are obviously relaxed when $m_{\tilde{\chi}^{0}_{1}}$ is above $100$ GeV
1501.07110 ; 1803.02762 .

The mixing effects in the VL electroweak sector are described by the mass
matrix $B$ in Eq.(77), which is similar to the higgsino-bino benchmark
scenario. See the reduced matrix A with wino decoupled in Eq.(77). We can
infer the implications of LHC limits above to the parameter space of
$M_{L}-M_{N}$ by introducing a ratio of signal strengths $\hat{\mu}$. For
example, for the process
$pp\rightarrow\tilde{\chi}^{\pm}_{E}\tilde{\chi}^{0}_{2}+$ jets $+$ leptons
1501.07110 ; 1803.02762  we have

$\displaystyle{}\hat{\mu}=\frac{\sigma(pp\rightarrow\tilde{\chi}^{\pm}_{E}\tilde{\chi}^{0}_{2}+\cdots)}{\sigma(pp\rightarrow\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{2}+\cdots)}\cdot\frac{\text{Br}(\tilde{\chi}^{\pm}_{E}\rightarrow
W^{\pm}\tilde{\chi}^{0}_{1})}{\text{Br}(\tilde{\chi}^{\pm}_{1}\rightarrow
W^{\pm}\tilde{\chi}^{0}_{1})}\cdot\frac{\text{Br}(\tilde{\chi}^{0}_{2}\rightarrow
Z\tilde{\chi}^{0}_{1})}{\text{Br}(\tilde{\chi}^{0}_{2}\rightarrow
Z\tilde{\chi}^{0}_{1})},$ (55)

where $\tilde{\chi}^{0}_{1}$ in the numerators (denominators) refers to the
lightest neutral fermion in the VL (higgsino-bino) sector. In the parameter
regions with $M_{N}<M_{L}$, where $\tilde{\chi}^{0}_{1}\sim\tilde{N}^{0}$,
$\tilde{\chi}^{0}_{2}\sim\tilde{\eta}^{0}$ and
$\tilde{\chi}^{\pm}_{E}\sim\tilde{E}^{\pm}$, the number of events for
$\tilde{\chi}^{\pm}_{E}\tilde{\chi}^{0}_{2}$ pair can be of same order as that
of $\tilde{H}_{u,d}^{\pm}\tilde{B}^{0}$. So the pattern of LHC limits
projected to the plane of $M_{N}-M_{L}$ is similar to Fig.8(d) in
ref.1803.02762 . In the parameter regions with $M_{N}>M_{L}$, where
$\tilde{\chi}^{0}_{1}\sim\tilde{\eta}^{0}$,
$\tilde{\chi}^{0}_{2}\sim\tilde{N}^{0}$, and
$\tilde{\chi}^{\pm}_{E}\sim\tilde{E}^{\pm}$, the mass splitting between
$\tilde{E}^{\pm}$ and $\tilde{\chi}^{0}_{1}$ is small similar to the case of
Higgsino-like $\tilde{\chi}^{\pm}_{1}$ and $\tilde{\chi}^{0}_{1}$. In this
situation, the decays 0104115  of charged fermion
$\tilde{\chi}^{\pm}_{E}\rightarrow\tilde{\chi}^{0}_{1}W^{*\pm}\rightarrow\tilde{\chi}^{0}_{1}\cdots$
are rather sensitive to the mass splitting.

##  VI Conclusion

In this paper, we have studied the minimal VL model motivated by the grand
unification. The key feature in such model is that the VL electroweak sector
couples to the Higgs doublets rather than mixes with the SM leptons.
Therefore, they can play important roles both in the phenomenologies of Higgs
physics and dark matter.

For the Higgs physics, we have used the LHC bounds on the Higgs coupling
constants to constrain the vevs in the VL electroweak sector, the magnitude of
which are found to be small. Consequently, the magnitudes of the mixing
effects between the Higgs doublets and the VL electroweak sector controlled by
the vevs are small as well, leading to small deviation in the Higgs self
coupling from the MSSM expectation. Moreover, we used the observed Higgs mass
to constrain the other model parameters in the VL sector. The fit reveals that
both large Yukawa coupling constant $k\sim 1.1$ and large mass splitting
between scalar and fermion masses in the VL sector are required.

For the dark matter phenomenology, w have verified that the large Yukawa
coupling $k$ excludes the possibility of VL dark matter. For it gives rise to
too large dark matter coupling to Higgs to evade the Xenon1T limit.
Nevertheless, they are still useful to reduce the tension on the neutralino
dark matter by relaxing the constraint from Higgs mass.

Finally, in order to be complete we briefly discussed the constraints on the
VL electroweak fermions at the LHC. Unlike in the 4-th generation leptons
which directly mix with SM leptons, the VL electroweak fermions can be either
produced via the decays of charginos and neutralinos, or they imitate the
electroweak productions of charginos and neutralinos. The prospect of the
later class at the high luminosity-LHC will be explored elsewhere Xu .

###### Acknowledgements.

We would like to thank P. Fayet for correspondence and H. Wu for a numerical
check in the case of neutralino dark matter. This research is supported by the
National Natural Science Foundation of China under Grant No.11775039, the
Chinese Scholarship Council, and the Fundamental Research Funds for the
Central Universities at Chongqing University under Grant No.cqu2017hbrc1B05.

##  Appendix A Scalar Mass Matrix

###  A.1 CP-even Scalar

In the basis
$\phi_{S}^{T}=(H^{0}_{uR},H^{0}_{dR},\eta_{R},\bar{\eta}_{R},N_{R})$, the
matrix elements for symmetric mass matrix squared $\mathcal{M}_{S}^{2}$ of CP-
even scalars in the Lagrangian

$\displaystyle{}\mathcal{L}\supset\frac{1}{2}\phi_{S}^{T,i}\mathcal{M}^{2}_{S,ij}\phi_{S}^{j}$
(56)

are as follows,

$\displaystyle{}\mathcal{M}^{2}_{S,11}$ $\displaystyle=$
$\displaystyle\mu^{2}+m^{2}_{H_{u}}+k^{2}(n^{2}+\bar{v}^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{S,12}$ $\displaystyle=$ $\displaystyle
b-khv_{L}\bar{v}_{L}$ $\displaystyle\mathcal{M}^{2}_{S,13}$ $\displaystyle=$
$\displaystyle-hn\mu-kh\bar{v}_{L}\upsilon_{d}+kM_{L}n$
$\displaystyle\mathcal{M}^{2}_{S,14}$ $\displaystyle=$ $\displaystyle
2k^{2}\upsilon_{u}\bar{v}_{L}+k(M_{N}n+h\upsilon_{d}v_{L})-kA_{k}n$
$\displaystyle\mathcal{M}^{2}_{S,15}$ $\displaystyle=$ $\displaystyle
2k^{2}n\upsilon_{u}-hv_{L}\mu-kM_{N}\bar{v}_{L}+kM_{L}v_{L}-kA_{k}\bar{v}_{L}$
$\displaystyle\mathcal{M}^{2}_{S,22}$ $\displaystyle=$
$\displaystyle\mu^{2}+m^{2}_{H_{d}}+h^{2}(n^{2}+v^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{S,23}$ $\displaystyle=$ $\displaystyle
2h^{2}\upsilon_{d}v_{L}+h(M_{N}n-k\upsilon_{u}\bar{v}_{L})-hA_{h}n$
$\displaystyle\mathcal{M}^{2}_{S,24}$ $\displaystyle=$ $\displaystyle kn\mu-
kh\upsilon_{u}v_{L}-hM_{L}n$ $\displaystyle\mathcal{M}^{2}_{S,25}$
$\displaystyle=$ $\displaystyle
2h^{2}n\upsilon_{d}+k\bar{v}_{L}\mu+hM_{N}v_{L}+hM_{L}\bar{v}_{L}+hA_{k}v_{L}$
$\displaystyle\mathcal{M}^{2}_{S,33}$ $\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{L}+h^{2}(n^{2}+\upsilon_{d}^{2})$
$\displaystyle\mathcal{M}^{2}_{S,34}$ $\displaystyle=$ $\displaystyle-
kh\upsilon_{u}\upsilon_{d}$ $\displaystyle\mathcal{M}^{2}_{S,35}$
$\displaystyle=$ $\displaystyle
2h^{2}nv_{L}-h\upsilon_{u}\mu+hM_{N}\upsilon_{d}+kM_{L}\upsilon_{u}+hA_{h}\upsilon_{d}$
$\displaystyle\mathcal{M}^{2}_{S,44}$ $\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{\bar{L}}+k^{2}(n^{2}+\upsilon^{2}_{u})$
$\displaystyle\mathcal{M}^{2}_{S,45}$ $\displaystyle=$ $\displaystyle
2k^{2}n\bar{v}_{L}+k\upsilon_{d}\mu-
kM_{N}\upsilon_{u}-hM_{L}\upsilon_{d}-kA_{k}\upsilon_{u}$
$\displaystyle\mathcal{M}^{2}_{S,55}$ $\displaystyle=$ $\displaystyle
M^{2}_{N}+m^{2}_{N}+h^{2}(v^{2}_{L}+\upsilon^{2}_{d})+k^{2}(\bar{v}^{2}_{L}+\upsilon^{2}_{u})$
(57)

where $D$-term contributions have been neglected. After diagonalizing the
matrix $\mathcal{M}_{S}^{2}$, we obtain five physical neutral scalars, one of
which severs as the SM-like Higgs boson $h$ with mass $125$ GeV Higgsmass1 ;
Higgsmass2 .

###  A.2 CP-odd Scalar

In the basis $\phi_{A}^{T}=(H_{uI},H_{dI},\eta_{I},\bar{\eta}_{I},N_{I})$ the
matrix elements for CP-odd scalars in the Lagrangian

$\displaystyle{}\mathcal{L}\supset\frac{1}{2}\phi_{A}^{T,i}\mathcal{M}^{2}_{A,ij}\phi^{j}_{A}$
(58)

are given by,

$\displaystyle{}\mathcal{M}^{2}_{A,11}$ $\displaystyle=$
$\displaystyle\mu^{2}+m^{2}_{H_{u}}+k^{2}(n^{2}+\bar{v}^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{A,12}$ $\displaystyle=$ $\displaystyle
b-khv_{L}\bar{v}_{L}$ $\displaystyle\mathcal{M}^{2}_{A,13}$ $\displaystyle=$
$\displaystyle-hn\mu-kh\bar{v}_{L}\upsilon_{d}+kM_{L}n$
$\displaystyle\mathcal{M}^{2}_{A,14}$ $\displaystyle=$
$\displaystyle-k(M_{N}n+h\upsilon_{d}v_{L})+kA_{k}n$
$\displaystyle\mathcal{M}^{2}_{A,15}$ $\displaystyle=$ $\displaystyle-
hv_{L}\mu-kM_{N}\bar{v}_{L}-kM_{L}v_{L}+kA_{k}\bar{v}_{L}$
$\displaystyle\mathcal{M}^{2}_{A,22}$ $\displaystyle=$
$\displaystyle\mu^{2}+m^{2}_{H_{d}}+h^{2}(n^{2}+v^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{A,23}$ $\displaystyle=$
$\displaystyle-h(M_{N}n-k\upsilon_{u}\bar{v}_{L})-hA_{h}n$
$\displaystyle\mathcal{M}^{2}_{A,24}$ $\displaystyle=$ $\displaystyle kn\mu-
kh\upsilon_{u}v_{L}-hM_{L}n$ $\displaystyle\mathcal{M}^{2}_{A,25}$
$\displaystyle=$
$\displaystyle+k\bar{v}_{L}\mu+hM_{N}v_{L}-hM_{L}\bar{v}_{L}-hA_{k}v_{L}$
$\displaystyle\mathcal{M}^{2}_{A,33}$ $\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{L}+h^{2}(n^{2}+\upsilon_{d}^{2})$
$\displaystyle\mathcal{M}^{2}_{A,34}$ $\displaystyle=$ $\displaystyle-
kh\upsilon_{u}\upsilon_{d}$ $\displaystyle\mathcal{M}^{2}_{A,35}$
$\displaystyle=$ $\displaystyle
h\upsilon_{u}\mu+hM_{N}\upsilon_{d}+kM_{L}\upsilon_{u}-hA_{h}\upsilon_{d}$
$\displaystyle\mathcal{M}^{2}_{A,44}$ $\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{\bar{L}}+k^{2}(n^{2}+\upsilon^{2}_{u})$
$\displaystyle\mathcal{M}^{2}_{A,45}$ $\displaystyle=$
$\displaystyle-k\upsilon_{d}\mu-
kM_{N}\upsilon_{u}-hM_{L}\upsilon_{d}+kA_{k}\upsilon_{u}$
$\displaystyle\mathcal{M}^{2}_{A,55}$ $\displaystyle=$ $\displaystyle
M^{2}_{N}+m^{2}_{N}+h^{2}(v^{2}_{L}+\upsilon^{2}_{d})+k^{2}(\bar{v}^{2}_{L}+\upsilon^{2}_{u})$

where the $D$-term contributions have been neglected. Under the decoupling
limit $v_{L}=\bar{v}_{L}=0$ and $n=0$, scalar $\eta_{I}$, $\bar{\eta}_{I}$ and
$N_{I}$ decouple from $H_{uI}$ and $H_{dI}$, which results in the well known
result $\text{Det}\mathcal{M}^{2}_{A}=0$.

###  A.3 CP-charged Scalar

In the basis $\phi_{c}^{T}=(H^{+}_{u},H^{-*}_{d},E^{+},E^{-*})$ the matrix
elements for charged scalars in the Lagrangian

$\displaystyle{}\mathcal{L}\supset\phi_{c}^{*T,i}\mathcal{M}^{2}_{C,ij}\phi^{j}_{c}$
(60)

read as

$\displaystyle{}\mathcal{M}^{2}_{C,11}$ $\displaystyle=$
$\displaystyle\mu^{2}+m^{2}_{H_{u}}+k^{2}(n^{2}+\bar{v}^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{C,12}$ $\displaystyle=$ $\displaystyle-
khv_{L}\bar{v}_{L}$ $\displaystyle\mathcal{M}^{2}_{C,13}$ $\displaystyle=$
$\displaystyle-hn\mu-kh\bar{v}_{L}\upsilon_{d}+kM_{L}n$
$\displaystyle\mathcal{M}^{2}_{C,14}$ $\displaystyle=$ $\displaystyle
k(M_{N}n-h\upsilon_{d}v_{L})+kA_{k}n$ $\displaystyle\mathcal{M}^{2}_{C,22}$
$\displaystyle=$ $\displaystyle\mu^{2}+m^{2}_{H_{d}}+h^{2}(n^{2}+v^{2}_{L})$
$\displaystyle\mathcal{M}^{2}_{C,23}$ $\displaystyle=$
$\displaystyle-h(M_{N}n-k\upsilon_{u}\bar{v}_{L})-hA_{h}n$
$\displaystyle\mathcal{M}^{2}_{C,24}$ $\displaystyle=$ $\displaystyle kn\mu-
kh\upsilon_{u}v_{L}-hM_{L}n$ $\displaystyle\mathcal{M}^{2}_{C,33}$
$\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{L}+h^{2}(n^{2}+\upsilon_{d}^{2})$
$\displaystyle\mathcal{M}^{2}_{C,34}$ $\displaystyle=$ $\displaystyle-
kh\upsilon_{u}\upsilon_{d}$ $\displaystyle\mathcal{M}^{2}_{C,44}$
$\displaystyle=$ $\displaystyle
M^{2}_{L}+m^{2}_{\bar{L}}+k^{2}(n^{2}+\upsilon^{2}_{u})$ (61)

Under the decoupling limit $v_{L}=\bar{v}_{L}=0$ and $n=0$, scalar $E^{+}$ and
$E^{-*}$ decouple from the others, due to which one recovers the relation
$\text{Det}\mathcal{M}^{2}_{C}=0$.

##  Appendix B Fermion Mass Matrix

###  B.1 Charged Fermions

In the basis
$\chi^{T}_{\pm}=(\tilde{W^{+}},\tilde{H_{u}^{+}},\tilde{E^{+}},\tilde{W^{-}},\tilde{H^{-}_{d}},\tilde{E^{-}})$
the mass matrix in
$\mathcal{L}\supset-\frac{1}{2}\chi_{\pm}^{T,i}\mathcal{M}_{\chi^{\pm},ij}\chi_{\pm}^{j}$
is given by,

$\displaystyle{}\mathcal{M}_{\chi^{\pm},ij}=\left(\begin{array}[]{cc}0&X^{T}\\\
X&0\\\ \end{array}\right)$ (64)

with

$\displaystyle{}X=\left(\begin{array}[]{ccc}M_{2}&\sqrt{2(1-\delta)}s_{\beta}M_{W}&\sqrt{2\delta}s_{\gamma}M_{W}\\\
\sqrt{2(1-\delta)}c_{\beta}M_{W}&\mu&-hn\\\
\sqrt{2\delta}c_{\gamma}M_{W}&kn&M_{L}\\\ \end{array}\right)$ (68)

where $s_{\beta}=\sin\beta$, $c_{\beta}=\cos\beta$, $s_{\gamma}=\sin\gamma$
and $c_{\gamma}=\cos\gamma$.

###  B.2 Neutral Fermions

In the basis
$\psi_{0}^{T}=(\tilde{B}^{0},\tilde{W}^{0},\tilde{H^{0}_{d}},\tilde{H^{0}_{u}},\tilde{\bar{\eta}},\tilde{\eta},\tilde{N})$
the mass matrix in
$\mathcal{L}\supset-\frac{1}{2}\psi_{0}^{T,i}\mathcal{M}_{\chi,ij}\psi_{0}^{j}$
reads as,

$\displaystyle{}M_{\chi}=\left(\begin{array}[]{cc}A&C\\\ &B\\\
\end{array}\right),$ (72)

with

$\displaystyle{}A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}M_{1}&0&-(1-\delta)M_{Z}s_{W}c_{\beta}&(1-\delta)M_{Z}s_{W}s_{\beta}\\\
&M_{2}&(1-\delta)M_{Z}c_{W}c_{\beta}&-(1-\delta)M_{Z}c_{W}s_{\beta}\\\
&*&0&-\mu\\\ &*&*&0\end{array}\right),$ (77) $\displaystyle B$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&-M_{L}&-k\upsilon_{u}\\\
&0&h\upsilon_{d}\\\ &*&-M_{N}\\\ \end{array}\right),$ (81) $\displaystyle C$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}-\delta
M_{Z}s_{W}c_{\gamma}&\delta M_{Z}s_{W}s_{\gamma}&0\\\ \delta
M_{Z}c_{W}c_{\gamma}&-\delta M_{Z}c_{W}s_{\gamma}&0\\\ 0&hn&hv_{L}\\\
-kn&0&-k\bar{v}_{L}\\\ \end{array}\right).$ (86)

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