# Parameter Efficient Tuning Allows Scalable Personalization of LLMs for Text
Entry: A Case Study on Abbreviation Expansion

Katrin Tomanek, Shanqing Cai, Subhashini Venugopalan
Google LLC

###### Abstract

Abbreviation expansion is a strategy used to speed up communication by
limiting the amount of typing and using a language model to suggest
expansions. Here we look at personalizing a Large Language Model’s (LLM)
suggestions based on prior conversations to enhance the relevance of
predictions, particularly when the user data is small ($\approx$1000 samples).
Specifically, we compare fine-tuning, prompt-tuning, and retrieval augmented
generation of expanded text suggestions for abbreviated inputs. Our case study
with a deployed 8B parameter LLM on a real user living with ALS, and
experiments on movie character personalization indicates that (1)
customization may be necessary in some scenarios and prompt-tuning generalizes
well to those, (2) fine-tuning on in-domain data (with as few as 600 samples)
still shows some gains, however (3) retrieval augmented few-shot selection
also outperforms fine-tuning. (4) Parameter efficient tuning allows for
efficient and scalable personalization. For prompt-tuning, we also find that
initializing the learned “soft-prompts” to user relevant concept tokens leads
to higher accuracy than random initialization.

Parameter Efficient Tuning Allows Scalable Personalization of LLMs for Text
Entry: A Case Study on Abbreviation Expansion

Katrin Tomanek, Shanqing Cai, Subhashini Venugopalan Google LLC

##  1 Introduction

Language models have long been used to reduce keystrokes and aid text entry in
smart keyboards. This work looks at models for such keyboard applications in
Augmentative and Alternative Communication (AAC) devices, in particular those
for users with severe motor impairments, e.g., people living with amyotrophic
lateral sclerosis (ALS) who communicate through eye-gaze typing. Recent
advances in the generative capabilities of large language models (LLMs) can
help significantly accelerate communication for such users. Prior studies
Adhikary et al. (2021); Cai et al. (2022); Shen et al. (2022) proposed
techniques for abbreviation expansion, where a user types short keywords or
abbreviated phrases consisting of the initial letter of each word and an LLM
is used to generate the fully-expanded sentence. Including the conversation
context Wisenburn and Higginbotham (2008); Gorman et al. (2021) was shown to
further improve the accuracy of the predictions. In this work we explore
personalization, another dimension that can improve the relevance of the
predictions to fit a user’s vocabulary and language style.

In many real-world applications personalization plays an important role in
enhancing the relevance of the suggested options and the quality of the user
experience Valencia et al. (2023). However, very little data is available to
adapt a model to a given user, and larger models increase the risk of
overfitting. Additionally, it remains unclear how to scale the approach to
multiple users given the high cost of LLM checkpoint storage and serving.

With these challenges in mind, our work evaluates three approaches to
personalizing LLMs for abbreviation expansion as used by eye-gaze typers.
Specifically we consider a pre-trained decoder-only LLM tuned for dialog
Roller et al. (2020); Thoppilan et al. (2022). We further fine-tune the model
on the abbreviation expansion task on data derived from dialog datasets. We
then compare personalizing this fine-tuned LLM on user-specific data via (1)
fine-tuning the entire model, (2) augmenting the LLM’s context by retrieving
similar conversations from the user’s history, and (3) parameter efficient
prompt-tuning Lester et al. (2021). Overall, prompt-tuning performed best and
retrieval augmented in-context learning (RA-ICL) also outperformed fine-
tuning.

##  2 Related Work

###  2.1 Language models for text-entry.

Using language models to expand abbreviated inputs for text-entry has been
well studied and different schemes of abbreviation have been proposed such as,
using just context words Demasco and McCoy (1992), discarding vowels Shieber
and Nelken (2007), and additionally omitting repeated consonants Willis et al.
(2005), flexible letter saving schemes Adhikary et al. (2021); Gorman et al.
(2021), and expanding from a bag of words Shen et al. (2022). Our study
focuses on abbreviation expansion used by eye-gaze typers living with severe
motor impairments. Given our goal to significantly reduce the number of
keystrokes, we consider a form of word-initial abbreviation similar to Cai et
al. (2022) where just the initial characters of the words are typed and an LLM
predicts the full sentence. The current study focuses on personalizing such a
model to a user, which has been less studied.

###  2.2 LLM prompt engineering.

LLMs have shown remarkable capabilities in understanding and performing tasks
with few-shot Brown et al. (2020) examples. However, the tokenization used in
LLMs makes our task of generating expansions from single characters somewhat
hard for the models. Due to this reason and to enable personalization, we
focus on Parameter Efficient Fine-Tuning (PEFT) Lester et al. (2021), and
retrieval augmented generation (RAG) Mialon et al. (2023). PEFT learns a small
set of additional parameters while keeping the weights of the original LLM
frozen. Many PEFT methods have been proposed in recent years. In case of
adapters Houlsby et al. (2019) and Low-Rank Optimization (LoRA) Hu et al.
(2021) these parameters are interspersed at different transformer layers of
the model. Other methods such as, Prompt-tuning Lester et al. (2021), Prefix-
tuning Li and Liang (2021), and P-tuning Liu et al. (2021) restrict the
parameters to the input prompt tokens. We use prompt-tuning Lester et al.
(2021) which append parameters to the token embeddings. We also compare this
to retrieval augmentation for ICL Rubin et al. (2022) where a dense retriever
is used to select relevant data point(s) that are then added as context to a
generative answering model. While most RAG studies Mialon et al. (2023) train
the retriever or the generator, we keep both of these pre-trained models
frozen. Specifically, we use the retrieved context to create more relevant
few-shot examples specific to the input query.

##  3 Tuning and personalization

Our broad approach consists of taking a pre-trained LLM, performing supervised
fine-tuning for the abbreviation expansion task, and then personalizing the
model on user data by means of further fine-tuning, prompt-tuning, or
retrieval augmented in-context few-shot generation. For the pre-trained model,
we start with an 8B parameter decoder-only LLM. This model is pre-trained on
the C4 dataset Raffel et al. (2019) and tuned for dialogs Roller et al.
(2020); Thoppilan et al. (2022). We then fine-tune it further for abbreviation
expansion on sentences from conversations and associated word-initial
abbreviated text. We follow prior works Cai et al. (2022) and experiment with
different learning rates, and use a constant rate during fine-tuning and
select the best based on a validation set. We refer to this as the base-AE
model. We explore 3 strategies for personalization.

###  3.1 Fine-tuning on user data.

We follow the same fine-tuning recipe on user data as with the base-AE model.
The tuning itself is fast since the amount of user data is small, and we
avoided overfitting by monitoring performance on the validation set. We
experimented with learning rates 1e-5, 1e-6, and 5e-5 and found 5e-5 to work
best (see App. Tab. 6).

###  3.2 Few-shot and Retrieval Augmented In-Context Learning (RA-ICL)

Another way to personalize an LLM is to provide it with few-shot examples to
allow for in-context learning (ICL). Performance with ICL can vary
significantly with few-shot examples Zhao et al. (2021). Hence, in addition to
typical few-shot examples, we also investigate a retrieval-augmented few-shot
setup. This is similar to works that retrieve from databases to augment LLMs
Mialon et al. (2023) but we use existing pre-trained models for retrieving and
generating, and keep them frozen. For the retriever, we use a pre-trained 11B
Sentence-T5 Ni et al. (2022) and generate embeddings of the abbreviated inputs
from the user conversations. Given a new input, we embed it and use Euclidean
distance to retrieve the nearest neighbor queries and the corresponding
expansions. We use this retrieved context to create relevant, query-specific
few-shot examples with which we prompt the LLM.

###  3.3 Prompt-tuning

We also investigate prompt-tuning Lester et al. (2021) for personalization.
The basic idea is to extend few-shot prompting and use substantially more in-
context examples to learn “soft-prompts” in the input embedding layer
specifically suited for the task at hand. We choose the length of the soft
prompt and initialize the tokens. For tuning, we correspondingly add new
learnable parameters to the model’s embedding matrix that are updated using
back propagation, keeping the original LLM weights frozen. The number of
learned parameters is a product of the length of the soft-prompt and
dimensions of the embedding weights. The learned soft-prompts are saved and
passed along with each user query to the LLM during inference. This approach
allows a single LLM to be served, and the soft-prompt to be swapped for
different users (see Sec. 6). The soft-prompts themselves can be tuned on
varying amounts of data, and are effective in low data settings Lester et al.
(2021). We train with a warm-up learning rate schedule with 1000 warm up steps
to a peak of 0.1 followed by linear decay. We use small batch sizes of 16 for
training and limit training to 20k steps. We experiment with different prompt
lengths and initialization strategies, and choose the best checkpoints based
on validation set accuracy.

##  4 Dataset

###  4.1 Abbreviation Expansion Base Model

To fine-tune the LLM for the abbreviation expansion task, we need pairs of
abbreviated phrases and the full expanded text. We use the data from Cai et
al. (2022) where they prepare paired sentences and abbreviated inputs from
four dialog datasets: crowd-sourced Turk Dialogues Vertanen (2017),
DailyDialog Li et al. (2017), the Cornell Movie Dialogues Danescu-Niculescu-
Mizil and Lee (2011) from movie scripts, and Turk AAC dataset Vertanen and
Kristensson (2011) of conversations collected with AAC users in mind. The
model fine-tuning is done with a constant low-learning rate (0.01) using the
AdaFactor optimizer Shazeer and Stern (2018) on over 237,000 examples derived
from the dialog datasets.

###  4.2 Personalization Dataset

A model trained on generic dialog datasets may not fit the communication needs
of all in terms of preserving their style, and vocabulary including proper
nouns. Our work is motivated to increase the autonomy and self-expression of
AAC users with motor and speech impairments and deploy our abbreviation
expansion model for their daily usage. This is also a case where a generic
models’ training data is also lacking in terms of conversations around
caregiving and health. Hence, our personalization dataset was collected from a
person living with ALS with informed consent from the user and the
conversation partners. They use eye-gaze text entry for everyday
communication. They type on a virtual keyboard into the text editor of a text-
to-speech (TTS) software and activate the audio to "speak" out the contents.
Private and sensitive content was redacted prior to obtaining the data for
research. The data was partitioned chronologically, and repetition was removed
from the validation and test portions resulting in 630 (train), 194 (val.) and
224 (test) samples.

###  4.3 Movie character personalization

Outside of the real deployment scenario, we also examined other conversation
datasets where personalization can be studied without affecting user privacy.
Characters in movies and TV series tend to have certain quirks and
personalities and make for a great test bed for evaluating personalization of
spoken dialogues. Thus, to evaluate the need for customization and scalability
of the approach, we performed additional experiments on conversations from the
Cornell Movie Dialogs dataset Danescu-Niculescu-Mizil and Lee (2011) test set.
For our experiments, we selected 10 movies with very high ratings (with
atleast 5k votes on ImDb). From each movie, we chose 1 character and all their
conversations from the movie for personalization. Each character had over a
hundred conversations in the movie (range 104 to 344, with a mean of 198.4 and
median of 209 conversations). Similar to our AAC personalization dataset we
did a time-based split of the data to get train, val., and test splits.

##  5 Experiments and Results

Experimental setup. For all experiments, we sample 128 responses from the
model with temperature 1.0, sort based on frequency of predictions and select
the top-5. We report results on the average (and $\pm$ std. dev.) of 3 runs
unless specified otherwise. The metrics we use are Accuracy to measure exact
match of the full sentence expansion, and BLEU score Papineni et al. (2002) to
consider partial credit, both measured on the top-5 predictions (noted as @5).

###  5.1 Prompt-tuning is best for Personalization

Table 1 compares the performance of the different personalization approaches
on the real user data. We note that the base-AE model achieved a top-5
accuracy of 68.3% on the abbreviation expansion test set, however from Tab. 1
we can see that it only gets an accuracy of 22.5% on the user personalization
test set highlighting the difference between the user data distribution and
the training distribution, and making a strong case for personalization for
AAC users. Fine-tuning on user data helps, and retrieval for ICL does even
better, however prompt-tuning results in the best performance.

Model | personalized | Accuracy@5 | BLEU@5
---|---|---|---
base-AE | $\times$ | 22.5 | 31.8
ICL | $\checkmark$ | 22.8 | 34.9
Fine-tuned | $\checkmark$ | 26.5 | 34.3
RA-ICL | $\checkmark$ | 30.3 | 39.1
Prompt-tuned | $\checkmark$ | 38.8 | 47.5

Table 1: Accuracy (exact-match of full sentence) and BLEU score of top 5
predictions of the different approaches on the personalization test set.

###  5.2 Soft prompt initialization matters

We experimented with different soft-prompt lengths, learning rates, and soft-
prompt initialization strategies. We tried soft-prompt lengths of 10, 25, and
100 tokens all initialized randomly. Recall that increasing the prompt lengths
increases the number of learned parameters. In our case, we found higher
prompt lengths led to more training instabilities. We found a length of 10 to
work best. Fixing the prompt length as 10, we experimented with learning rates
of 0.1, 0.2, and 0.3 and found 0.1 to work best (in App. Tab. 7).

Soft-prompt Initialization | Accuracy@5 | BLEU@5
---|---|---
Random | 32.7 $\pm 3.2$ | 43.6 $\pm 2.3$
LLM vocab. sampled | 33.9 $\pm 0.4$ | 43.2 $\pm 1.8$
User vocab. sampled | 32.6 $\pm 1.6$ | 41.0 $\pm 1.9$
User relevant concepts |  36.8 $\pm 1.9$ |  45.9  $\pm 1.4$
User concept antonyms |  36.4 $\pm 0.3$ |  46.2 $\pm 4.3$

Table 2: Initializing soft-prompts with proper nouns and concepts from the
user’s data performs best.

The thing that made the biggest difference though was the choice of
initialization for the soft-prompt token embeddings, which can be seen in
Table 2. We examined 5 strategies, (1) random initialization, (2) sampling
from the top 5k words in the LLM’s vocabulary, (3) sampling from the top 25
most common English words in the user’s vocabulary, (4) hand-selecting proper
names and concepts relevant to the user (e.g. ALS) and (5) selecting words
that are related but might be considered antonyms of the user concepts (e.g.
Parkinsons). We found the initialization that relied on the user concepts to
perform significantly better. Analogous to what is suggested in Lester et al.
(2021), perhaps these tokens are the ones the base model is most uncertain
about, and hence boosts their chance of appearing in the predictions when
prompt-tuned.

###  5.3 Fine-tuning hampers generalization

Fig. 1 slices performance of the models based on the length of the sentences.
The performance of all models degrade with increasing sentence length.
However, the fine-tuned model generalizes poorly compared to the base-AE model
in some cases (noticeable at lengths 5 and 6). This also highlights the
difficulty with fine-tuning large models on very small datasets.

Figure 1: Performance of the different approaches on conversation sentences of
different lengths. On longer sentences, the prompt-tuned (green) and base non-
personalized model (in blue) can outperform fine-tuning highlighting their
capacities to generalize to the long tail of complex sentences.

ICL (4-shot) strategy | Accuracy@5 | BLEU@5
---|---|---
Random 4-shot | 22.0 $\pm 0.9$ | 30.8 $\pm 1.4$
Hand-crafted 4-shot | 21.9 $\pm 0.5$ | 33.3 $\pm 0.7$
Retrieval augmented (RA-ICL) | 30.2 $\pm 0.3$ | 38.5 $\pm 0.6$

Table 3: Comparing different few-shot selection strategies. Retrieval
augmented ICL works best.

###  5.4 Retrieval augmented few-shots help

Table 3 presents results for in-context learning where 4-shot examples are
selected using different strategies: (1) random selection from the training
set, (2) hand-crafted examples containing proper names of people that the user
communicates with, and (3) retrieval-augmented few shots, where 4 nearest
neighbor examples (in the embedding space of abbreviations) are selected based
on each test query. RA-ICL outperforms other strategies by a large margin.

Movie-id | character-id | Character-Name | Non-personalized base-AE | Personalized (prompt-tuned) | Personalization
---|---|---|---|---|---
|  |  | Acc. @5 | BLEU @5 | Acc. @5 | BLEU @5 | rel. benefit (%)
m106 | u1612 | JACOB | 62.75 | 67.03 | 56.86 | 65.13 | -
m119 | u1799 | GEORGE | 50.00 | 59.18 | 56.25 | 62.46 | 13%
m126 | u1916 | ANDERTON | 44.12 | 55.66 | 38.24 | 55.74 | -
m140 | u2157 | BABE | 60.00 | 69.52 | 46.67 | 62.93 | -
m148 | u2299 | NANCY | 41.67 | 52.67 | 41.67 | 51.40 | -
m203 | u3105 | MICHAEL | 61.90 | 59.60 | 47.62 | 45.32 | -
m274 | u4099 | ABBY | 77.78 | 77.78 | 77.78 | 77.78 | -
m324 | u4866 | SONNY | 62.86 | 71.53 | 65.71 | 72.97 | 5%
m352 | u5310 | JACK | 50.00 | 59.18 | 56.25 | 62.46 | 13%
m565 | u8329 | JEANNE | 61.54 | 70.61 | 64.10 | 71.24 | 4%

Table 4: Performance comparison between Non-personalized base-AE and
Personalized (prompt-tuned) models on movie character personalization. LLMs
raise the bar on average performance indicating that customization may not
always be necessary on certain conversation categories, though some users
benefit from it. This is a contrast to the real AAC deployment scenario.

###  5.5 Customization is not always necessary

We also evaluated the prompt-tuning approach on the movie character
personalization dataset and report results in Table 4. We observe that: (1)
the base non-personalized model accuracies do seem to transfer reasonably well
indicating that customization may not be necessary for conversation types
similar to the training data distribution. (2) 4 of the 10 speakers still
benefit from personalization. (3) the proposed prompt-tuning approach offers a
way to serve the same end-point, while optionally choosing to personalize
results to some users.

Error Type | Abbreviation | Gold Expansion | Fine-tuned | Prompt-tuned
---|---|---|---|---
Unmatched Acronym | s i l t r |  sweet i love that robin | i love that robin | sweet i love that robin
Overfitting to names | g q d , r a m | great question dude , robin and mommy |  greg q day, robin and greg | good q doc, robin and mommy
Misses user style (often | w a d , o d y | what a dunce , okie dokie yall | what about daddy , okie dokie | what a day , okie dokie
when lacking context) | wipe and dry , ok thanks | we are done , ok day yall

Table 5: Examples of some observed categories of errors. Words that the model
misses are highlighted in blue in the Gold expansion, and errors in names are
marked in red. (Proper names have been changed to preserve anonymity)

###  5.6 Error Analysis

In Table 5 we share some examples of the categories of errors we observe
comparing the fine-tuned and prompt-tuned results. Our analysis of the
predictions show that the fine-tuned model tends to overfit to proper nouns in
the user’s training data, and often misses generating expansions for some of
the characters in the acronym. On sessions where there is not enough user
context, it can miss the user’s style (e.g. the word contraction “yall” in row
4 of Table 5 is less common in general text, but could be stylistic of a
user).

##  6 Discussion

###  6.1 LLM Blind-spots.

Abbreviation expansion may seem to be an easy task for current LLMs. However,
our work focuses on abbreviations motivated to help users with severe
disabilities, and hence pushes the limit of keystroke savings. Wherein, the
task depends on recognizing individual characters/alphabets. Interestingly, it
falls into what could be a "blind-spot" for the LLMs because the input
tokenization schemes - meant to overcome a discrete vocabulary - may fall
short in recognizing individual characters. This is now addressed practically
e.g. for generating text in JSON format Lamini (2023), using constrained
decoding and following Backus-Naur Form (BNF) grammar.

###  6.2 Data efficiency and scaling.

Another point of discussion is how personalization can be performed on a small
amount of data. Our experiments show that prompt-tuning leads to higher test
accuracy than fine-tuning in limited data settings. Fine-tuning the full LLM
for personalization not only generalizes poorly, but is also very expensive in
terms of storing the personalized model weights. Prompt-tuning on the other
hand only involves storing a very small set of weights (on the order of
thousands) which would make it not only possible but also convenient to store
these on users’ personal devices. This also makes the approach more scalable
since only a single model can be served, while clients can query it using
different personalized soft prompts. Further, querying a prompt-tuned model
incurs little additional inference latency, as the learned prompts and the
user input are provided simultaneously to the model during inference.

##  7 Conclusion

Our work presents a case study on personalizing LLMs for the task of
abbreviation expansion in the context of aiding eye-gaze typers with severe
motor and speech disabilities to communicate faster. We fine-tuned an LLM on
generic dialog data for the task and compared approaches to personalization
using limited user data. We examined fine-tuning, parameter-efficient prompt-
tuning, and retrieval augmented in-context learning, and find prompt-tuning to
be the most elegant method for personalization in terms of its performance as
well as its training data efficiency, small storage requirements, and ability
to scale. Further, initializing the soft-prompts with concepts and terms
relevant to the user resulted in better prompt-tuned personalized models.

## Limitations

The effectiveness of personalization on real usage is difficult to study,
since it deals with private and sensitive content. This difficulty is more
pronounced when working with people with disabilities. This limits our work to
a case study on real user data for personalization. Identifying interesting
techniques to collect realistic personalization datasets, perhaps synthetic,
can benefit the community significantly.

We also limit the extent of hyperparameter tuning, due to significant
computation resource consumption of experiments. Though we are able to take
advantage of settings shared in literature and open source code. Also, while
our abbreviation expansion study and models are limited to English, it will
likely translate well to languages with similar morphology, but that remains
to be studied. Our references to related work in this space may be limited and
further suggestions are welcome.

## Ethics and Societal Impact

Techniques that improve Augmentative and Alternative Communication (AAC)
applications can significantly enhance quality of life, increase independence
and social participation Caligari et al. (2013) of people living with
communication and motor disabilities.

A risk of abbreviation expansion is that, when the expansions are not exactly
the ones that the user desires, they may be tempted to choose a near similar
prediction leading to conveying content that may be less accurate,
misinterpreted, or reflecting biases and stereotypes of the underlying models.
While the goal of personalization is to mitigate these, some of the risks
still remain. Hence there is still a subtle risk of reducing speaker’s
autonomy and authentic self-expression which people e.g. with ALS Kane et al.
(2017) value highly. Another risk is that of frequent incorrect predictions if
personalization is poor for some users. This could increase effort required to
edit minor errors, and inadvertently increase fatigue.

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##  Appendix A Parameter selection

###  A.1 Fine-tuning learning rates

Fine-tuning learning rate | Accuracy@5 | BLEU@5
---|---|---
5e-5 | 26.8 | 34.3
1e-6 | 25.4 | 34.7
1e-5 | 23.7 | 31.6

Table 6: Comparing different learning rates for fine-tuning base-AE model on
personalization data. (val set).

###  A.2 Prompt-tuning learning rates

Prompt-tuning learning rate | Accuracy@5 | BLEU@5
---|---|---
0.1 | 35.7 | 45.6
0.2 | 31.7 | 41.7
0.3 | 30.8 | 39.9

Table 7: Comparing different learning rates for prompt-tuning base-AE model on
personalization data. soft prompt length of 10 and random initialization (val
set).

##  Appendix B Personalization Data

Our personalization dataset was collected with informed consent from a person
living with ALS over a period of five months from late 2021 to early 2022. We
refer to the person with ALS as "the user". The user used a Tobii (R) eye-
tracker and gaze-driven keyboard to enter text for daily communication. The
gaze-typed text was output as speech audio through text-to-speech (TTS)
software. The user had full control over when to start and stop data
collection. Private and sensitive content in the data was redacted by trained
human curators prior before we ingested the dataset for research.

The relevant data used for this study consists of text transcripts of the
user’s TTS output. We split the data into three non-overlapping splits along
the time axis in chronological order as train, validation and test, containing
630, 285, and 284 sentences, respectively. We filter the validation and test
split to preserve only the sentences with abbreviation length $\leq 10$,
leading to 194 and 224 sentences, respectively. No filtering is done on the
training split. As a result, the average abbreviation length in the train,
validation, and test splits are $6.91\pm 6.25$, $4.72\pm 2.39$, and $5.05\pm
2.74$, respectively ($\pm 1SD$). The sentences belong to 122, 69, and 72
sessions, respectively, each session being a continuous period of conversation
data collection. The percentages of proper nouns among the words were
$6.73\%$, $5.88\%$, and $8.61\%$ in the three splits, respectively.


# Spectrum of $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems in a chiral SU(3)
quark model

Du Wang  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 101408, China  Ke-Rang Song  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
101408, China  Wen-Ling Wang<EMAIL_ADDRESS>School of Physics,
Beihang University, Beijing 100191, China  Fei Huang<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 101408, China

###### Abstract

Inspired by the resonance $T_{cc}^{+}(3875)$ recently observed by the LHCb
Collaboration, we systematically explore the $S$\- and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems in a chiral SU(3)
quark model. The Hamiltonian contains the kinetic energy, the one-gluon-
exchange (OGE) potential, the confinement potential, and the one-boson-
exchange (OBE) potential stemming from the coupling of quark and chiral
fields. The Schrödinger equation is solved by use of the variational method
with the spacial trial wave functions chosen as Gaussian functions. It is
found that the lowest state has a mass $3879$ MeV, isospin and spin-parity
$IJ^{P}=01^{+}$, and quark constituent $cc\bar{u}\bar{d}$, in agreement with
the experimentally observed $T_{cc}^{+}(3875)$. This state is approximately at
the calculated $DD^{\ast}$ threshold, and has a root-mean-square radius about
$0.48$ fm. These demonstrates that the $T_{cc}^{+}(3875)$ can be accommodated
as a stable and compact tetraquark sate in the chiral SU(3) quark model. All
the other $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states lie about one
hundred to few hundreds MeV higher than the corresponding meson-meson
thresholds, and thus are not suggested to be candidates of stable and compact
tetraquark states due to their fall-apart decays to two mesons.

##  I Introduction

For long, we believe that hadrons can be divided into two categories, i.e.
meson consists of a pair of quark and antiquark ($q\bar{q}$) and baryon
consists of three quarks ($qqq$). However, in principle, the underlying theory
of strong interactions – quantum chromodynamics (QCD) – allows the existences
of any color-singlet states like tetraquarks, pentaquarks, and baryonium, etc.
In experiment, a large number of candidates of exotic states have been
reported in the last two decades after the observation of the $X(3872)$ by the
Belle Collaboration in 2003 Acosta et al. (2004). Theoretically, these
candidates have aroused significant research interest, and plenty of works
have been published to study the structures and properties of these possible
exotic states. See Refs. Hosaka et al. (2016); Ali et al. (2017); Guo et al.
(2018); Esposito et al. (2017); Lebed et al. (2017); Richard (2016); Chen et
al. (2016); Liu et al. (2019); Brambilla et al. (2020) for recent reviews of
the experimental and theoretical status of the investigations of exotic
states.

Recently, a doubly charmed exotic meson $T_{cc}^{+}(3875)$ was announced by
the LHCb Collaboration Aaij et al. (2022a, b). This state has spin-parity
$J^{P}=1^{+}$ and quark constituent $cc\bar{u}\bar{d}$. Its mass is reported
to be $M=3875.1$ MeV $+\delta m$ with $\delta m=-273\pm 61\pm 5^{+11}_{-14}$
keV, very close to the meson-meson threshold of $DD^{\ast}$. It has a very
small decay width, $\Gamma=410\pm 165\pm 43^{+18}_{-38}$ keV, and is so far
the longest living exotic state.

The $T_{cc}^{+}(3875)$ particle is of particular interest as it is really an
exotic state, unlike most of the reported candidates that have the same
quantum numbers as the traditional mesons or baryons and thus are not easy to
be clearly identified as exotic states. In literature, the $T_{cc}^{+}(3875)$
state has been investigated by use of various methods, e.g., quark models
Ebert et al. (2007); Lü et al. (2020); Wang et al. (2022); Vijande et al.
(2009); Meng et al. (2021); Deng and Zhu (2022), QCD sum rules Navarra et al.
(2007); Du et al. (2013); Agaev et al. (2022), lattice QCD Ikeda et al.
(2014); Junnarkar et al. (2019), and one-boson-exchange (OBE) model Li et al.
(2013), et al. Unfortunately, the conclusions from different works are not
consistent, even they are done within the same method. In the quark model
studies of Refs. Ebert et al. (2007); Lü et al. (2020); Wang et al. (2022), no
stable doubly-charmed tetraquark states were found because the lowest
$cc\bar{u}\bar{d}$ states were reported to be about $60-200$ MeV higher than
the thresholds of two open-charmed mesons for rearrangement decays. On the
contrary, in the quark model studies of Refs. Vijande et al. (2009); Meng et
al. (2021); Deng and Zhu (2022), opposite conclusions were drawn. In Ref.
Vijande et al. (2009), the state $cc\bar{u}\bar{d}$ with spin-parity
$J^{P}=1^{+}$ was reported to be a compact tetraquark state with a mass $65$
MeV below the $DD^{\ast}$ threshold. In Ref. Meng et al. (2021), the
tetraquark state $cc\bar{u}\bar{d}$ with $J^{P}=1^{+}$ was found to have a
mass $23$ MeV below the $DD^{\ast}$ threshold. In Ref. Deng and Zhu (2022),
the lowest $cc\bar{u}\bar{d}$ state was claimed to be a loosely bound
$DD^{\ast}$ molecular state with $0.34$ MeV binding energy. In the studies by
QCD sum rule, Refs. Navarra et al. (2007); Du et al. (2013) reported that the
calculated $cc\bar{u}\bar{d}$ state with $J^{P}=1^{+}$ are above the
$DD^{\ast}$ threshold, while Ref. Agaev et al. (2022) claimed that the
calculated mass $M=3868\pm 124$ MeV and width $\Gamma=489\pm 92$ keV for
$cc\bar{u}\bar{d}$ with $J^{P}=1^{+}$ agree well with the experiment. In the
lattice QCD study of Ref. Ikeda et al. (2014), neither bound nor resonance
states were reported, while in Ref. Junnarkar et al. (2019), a bound state
with binding energy $23$ MeV was found. In the OBE model study of Ref. Li et
al. (2013), the doubly charmed tetraquark state with $J^{P}=1^{+}$ was
suggested to be a $DD^{\ast}$ molecule.

The poor situation that diverging conclusions for the $cc\bar{u}\bar{d}$
tetraquark state were resulted from different theoretical works, regardless
whether they were done with the same or different theoretical methods, urges
upon us the importance and indispensable of further independent theoretical
analysis with reliable model ingredients for a better understanding of the
nature of the $cc\bar{u}\bar{d}$ tetraquark state.

In our previous work of Ref. Huang and Wang (2018), we have successfully
described the energies of octet and decuplet baryon ground states, the binding
energy of deuteron, the nucleon-nucleon ($NN$) scattering phase shifts and
mixing parameters for angular momentum up to $J=6$ within a chiral SU(3) quark
model. The Hamiltonian consists of the kinetic energy, the one-gluon-exchange
(OGE) potential, the confinement potential, and the one-boson-exchange (OBE)
potential stemming from the coupling of quark and chiral fields. It should be
mentioned that the work of Ref. Huang and Wang (2018) was the first and so far
the only one that reproduced the energies of octet and decuplet baryon ground
states and the experimental data of $NN$ scattering in a rather consistent
manner in a quark model. It solved the issue that the wave functions chosen
for single baryons are not the solutions of the given Hamiltonian in the
resonating-group-method (RGM) study of the baryon-baryon systems in
constituent quark models.

In the present work, we further extend the chiral SU(3) quark model employed
in Ref. Huang and Wang (2018) to explore the mass spectra of the $S$\- and
$P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems. The interactions
between the light antiquarks $\bar{q}$ and $\bar{q}^{\prime}$ are taken from
Ref. Huang and Wang (2018). The interactions associated with charm quark
consist of the OGE potential and the confinement potential, and the
corresponding parameters are fixed by a fit to the masses of known charmed
baryons and mesons. The total wave functions of $cc\bar{q}\bar{q}^{\prime}$
tetraquark systems are constructed as combinations of the individual wave
functions in the color, spin, flavor, and coordinate spaces, where the spacial
trial wave functions are chosen as Gaussian functions. The masses and
eigenvectors for the $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states are obtained by
solving the Schrödinger equation via the variational method.

This paper is organized as follows. In Sec. II, we introduce the Hamiltonian
employed in the chiral SU(3) quark model and the determination of the model
parameters. In Sec. III, we construct the spin, flavor, color, and spacial
wave functions and present the relevant matrix elements of interaction
operators for the doubly charmed tetraquark states. The numeric results of the
mass spectra of the $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$ tetraquark
states are shown in Sec. IV, where some discussions are presented as well.
Finally, in Sec. V, we give a brief summary.

##  II Hamiltonian

The Hamiltonian of the $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ system consists of the
masses of constituent quarks, the kinetic energy of the system, and the
potential between each pair of the constituent quarks, e.g.,

$H=\sum_{i=1}^{4}\left(m_{i}+T_{i}\right)-T_{G}+\sum_{1=i<j}^{4}\left(V_{ij}^{\text{conf}}+V_{ij}^{\text{OGE}}+V_{ij}^{\text{OBE}}\right),$
(1)

where $m_{i}$ and $T_{i}$ represent the mass and kinetic energy of the $i$th
constituent quark, respectively, and $T_{G}$ denotes the kinetic energy of the
center-of-mass motion of the $cc\bar{q}\bar{q}^{\prime}$ system,

$T_{i}=\frac{{\bm{p}}_{i}^{2}}{2m_{i}},\qquad
T_{G}=\frac{\left(\sum_{i=1}^{4}\bm{p}_{i}\right)^{2}}{2\sum_{i=1}^{4}m_{i}},$
(2)

with $\bm{p}_{i}$ being the three-momentum of the $i$th constituent quark. The
potential between each pair of constituent quarks is composed of three parts:
the phenomenological confinement potential $V_{ij}^{\text{conf}}$, the OGE
potential $V_{ij}^{\text{OGE}}$, and the OBE potential $V_{ij}^{\text{OBE}}$
stemming from the coupling of quark and chiral fields. The last one exists
only between a pair of light quarks (antiquarks) in a chiral SU(3) quark model
Huang and Wang (2018).

The confinement potential $V_{ij}^{\text{conf}}$ describes the long-range non-
perturbative QCD effects. In the present work, the linear type confinement
potential is adopted,

$V_{ij}^{\text{conf}}=-\bm{\lambda}_{i}^{c}\cdot\bm{\lambda}_{j}^{c}\left(a_{ij}r_{ij}+a_{ij}^{0}\right),$
(3)

where $\bm{\lambda}_{i}^{c}$ is the usual Gell-Mann matrix of the color SU(3)
group, and $a_{ij}$ and $a_{ij}^{0}$ are parameters of the confinement
strength and zero-point energy, respectively.

The OGE potential $V_{ij}^{\text{OGE}}$ describes the short-range perturbative
QCD effects. As usual, it can be written as

$\displaystyle V_{ij}^{\rm OGE}=$
$\displaystyle\;\frac{g_{i}g_{j}}{4}\bm{\lambda}_{i}^{c}\cdot\bm{\lambda}_{j}^{c}\left[\frac{1}{r_{ij}}-\frac{\mu_{ij}^{3}}{2}\frac{e^{-\mu_{ij}^{2}r_{ij}^{2}}}{\mu_{ij}r_{ij}}\left(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}\right.\right.$
$\displaystyle\left.\left.+\,\frac{4}{3}\frac{\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}}{m_{i}m_{j}}\right)\right]+V_{\text{ls}}^{\rm
OGE}(\bm{r}_{ij})+V_{\text{ten}}^{\rm OGE}(\bm{r}_{ij}),$ (4)

with

$\displaystyle V^{\rm OGE}_{\rm ls}({\bm{r}}_{ij})=$
$\displaystyle-\frac{g_{i}g_{j}}{4}\bm{\lambda}^{c}_{i}\cdot\bm{\lambda}^{c}_{j}\frac{m_{i}^{2}+m_{j}^{2}+4m_{i}m_{j}}{8m_{i}^{2}m_{j}^{2}}\frac{1}{r^{3}_{ij}}$
$\displaystyle\times\left[{\bm{L}\cdot\left(\bm{\sigma}_{i}+\bm{\sigma}_{j}\right)}\right],$
(5) $\displaystyle V_{\text{ten}}^{\rm OGE}(\bm{r}_{ij})=$
$\displaystyle-\frac{g_{i}g_{j}}{4}\bm{\lambda}_{i}^{c}\cdot\bm{\lambda}_{j}^{c}\frac{1}{4m_{i}m_{j}}\frac{1}{r_{ij}^{3}}$
$\displaystyle\times\left[3\left(\bm{\sigma}_{i}\cdot\hat{\bm{r}}_{ij}\right)\left(\bm{\sigma}_{j}\cdot\hat{\bm{r}}_{ij}\right)-\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\right],$
(6)

where $g_{i(j)}$ is the OGE coupling constant for the $i(j)$-th constituent
quark, and $\mu_{ij}$ is defined as
$\mu_{ij}\equiv\beta\frac{m_{i}m_{j}}{m_{i}+m_{j}}$ with $\beta$ being a model
parameter.

Note that in both confinement potential and OGE potential [cf. Eqs. (3)-(6)],
the Gell-Mann matrix $\bm{\lambda}^{c}$ for a quark should be replaced by
$-{\bm{\lambda}^{c}}^{*}$ for an antiquark.

In the chiral SU(3) quark model, the OBE potential is introduced in such a way
that the Lagrangian of the quark and chiral fields is invariant under the
chiral SU(3) transition Zhang et al. (1997); Huang and Zhang (2004); Huang et
al. (2004a); Huang and Wang (2018), which gives a natural explanation of the
relatively large constituent quark masses via the mechanism of spontaneous
chiral symmetry breaking. At the meanwhile, the Goldstone bosons obtain their
physical masses via the obvious chiral symmetry breaking caused by the tiny
current quark masses. The OBE potential provides the necessary medium- and
long-range attraction in light quark systems. In the $NN$ systems, it has been
shown that such attraction is rather important for describing the experimental
data Huang and Wang (2018).

The OBE potential between a pair of light quarks (antiquarks) reads

$V_{ij}^{\text{OBE}}=\sum_{a=0}^{8}V_{ij}^{\sigma_{a}}+\sum_{a=0}^{8}V_{ij}^{\pi_{a}},$
(7)

where the first and second terms represent the potential stemming from the
exchanges of the scalar nonet mesons and pseudoscalar nonet mesons,
respectively. The explicit expressions of $V_{ij}^{\sigma_{a}}$ and
$V_{ij}^{\pi_{a}}$ are

$\displaystyle V_{ij}^{\sigma_{a}}=$
$\displaystyle-C(g_{\mathrm{ch}},m^{\prime}_{\sigma_{a}},\Lambda)\,Y_{1}(m^{\prime}_{\sigma_{a}},\Lambda,r_{ij})\left(\lambda_{i}^{a}\lambda_{j}^{a}\right)$
$\displaystyle+V_{\mathrm{ls}}^{\sigma_{a}}(\bm{r}_{ij}),$ (8) $\displaystyle
V_{ij}^{\pi_{a}}=$
$\displaystyle~{}C(g_{\mathrm{ch}},m^{\prime}_{\pi_{a}},\Lambda)\frac{m_{\pi_{a}}^{\prime
2}p_{ij}^{a}}{48}Y_{3}(m^{\prime}_{\pi_{a}},\Lambda,r_{ij})\left(\lambda_{i}^{a}\lambda_{j}^{a}\right)$
$\displaystyle\times\left(\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\right)+V_{\mathrm{ten}}^{\pi_{a}}(\bm{r}_{ij}),$
(9)

with

$\displaystyle V_{\rm ls}^{\sigma_{a}}(\bm{r}_{ij})=$
$\displaystyle-C(g_{\mathrm{ch}},m_{\sigma_{a}}^{\prime},\Lambda)\frac{m_{\sigma_{a}}^{\prime
2}s_{ij}^{a}}{8}Z_{3}(m_{\sigma_{a}}^{\prime},\Lambda,r_{ij})$
$\displaystyle\times\left[\bm{L}\cdot\left(\bm{\sigma}_{i}+\bm{\sigma}_{j}\right)\right]\left(\lambda_{i}^{a}\lambda_{j}^{a}\right),$
(10) $\displaystyle V_{\rm ten}^{\pi_{a}}(\bm{r}_{ij})=$
$\displaystyle~{}C(g_{\mathrm{ch}},m_{\pi_{a}}^{\prime},\Lambda)\frac{m_{\pi_{a}}^{\prime
2}p_{ij}^{a}}{48}H_{3}(m_{\pi_{a}}^{\prime},\Lambda,r_{ij})$
$\displaystyle\times\left[3\left(\bm{\sigma}_{i}\cdot\hat{\bm{r}}_{ij}\right)\left(\bm{\sigma}_{j}\cdot\hat{\bm{r}}_{ij}\right)-\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\right]\left(\lambda_{i}^{a}\lambda_{j}^{a}\right),$
(11)

where

$\displaystyle C(g_{\mathrm{ch}},m,\Lambda)=$
$\displaystyle~{}\frac{g_{\mathrm{ch}}^{2}}{4\pi}\frac{\Lambda^{2}}{\Lambda^{2}-m^{2}}m,$
(12) $\displaystyle Y_{1}(m,\Lambda,r)=$
$\displaystyle~{}Y(mr)-\frac{\Lambda}{m}Y(\Lambda r),$ (13) $\displaystyle
Y_{3}(m,\Lambda,r)=$
$\displaystyle~{}Y(mr)-\left(\frac{\Lambda}{m}\right)^{3}Y(\Lambda r),$ (14)
$\displaystyle Z_{3}(m,\Lambda,r)=$
$\displaystyle~{}Z(mr)-\left(\frac{\Lambda}{m}\right)^{3}Z(\Lambda r),$ (15)
$\displaystyle H_{3}(m,\Lambda,r)=$
$\displaystyle~{}H(mr)-\left(\frac{\Lambda}{m}\right)^{3}H(\Lambda r),$ (16)
$\displaystyle Y(x)=$ $\displaystyle~{}\frac{1}{x}e^{-x},$ (17) $\displaystyle
Z(x)=$ $\displaystyle~{}\left(\frac{1}{x}+\frac{1}{x^{2}}\right)Y(x),$ (18)
$\displaystyle H(x)=$
$\displaystyle~{}\left(1+\frac{3}{x}+\frac{3}{x^{2}}\right)Y(x),$ (19)

and

$\displaystyle s_{ij}^{a}=$
$\displaystyle\left\\{\begin{array}[]{lccl}\dfrac{1}{m_{i}^{2}}+\dfrac{1}{m_{j}^{2}},&&&~{}(a=0,1,2,3,8)\\\\[12.0pt]
\dfrac{2}{m_{i}m_{j}},&&&~{}(a=4,5,6,7)\end{array}\right.$ (22) $\displaystyle
p_{ij}^{a}=$
$\displaystyle\left\\{\begin{array}[]{lcl}\dfrac{4}{m_{i}m_{j}},&&(a=0,1,2,3,8)\\\\[9.0pt]
\dfrac{\left(m_{i}+m_{j}\right)^{2}}{m_{i}^{2}m_{j}^{2}},&&(a=4,5,6,7)\end{array}\right.$
(25) $\displaystyle m_{\sigma_{a}}^{\prime}=$
$\displaystyle\left\\{\begin{array}[]{ll}m_{\sigma_{a}},&(a=0,1,2,3,8)\\\\[6.0pt]
\sqrt{m_{\sigma_{a}}^{2}-\left(m_{i}-m_{j}\right)^{2}},&(a=4,5,6,7)\end{array}\right.$
(28) $\displaystyle m_{\pi_{a}}^{\prime}=$
$\displaystyle\left\\{\begin{array}[]{ll}m_{\pi_{a}},&(a=0,1,2,3,8)\\\\[6.0pt]
\sqrt{m_{\pi_{a}}^{2}-\left(m_{i}-m_{j}\right)^{2}}.&(a=4,5,6,7)\end{array}\right.$
(31)

Here, $g_{\rm ch}$ is the quark and chiral field coupling constant, $\Lambda$
is the cutoff parameter indicating the chiral symmetry breaking scale, and
$m_{\pi_{a}}$ and $m_{\sigma_{a}}$ $(a=0,1,2,\cdots,8)$ represent the masses
of nonet pseudoscalar and nonet scalar mesons, respectively.

For pseudoscalar meson exchanges, the couplings of $\eta_{0}$ and $\eta_{8}$
are considered to give the physical $\eta$ and $\eta^{\prime}$ states:

$\displaystyle\left\\{\begin{array}[]{l}\eta=\eta_{8}\cos\theta-\eta_{0}\sin\theta,\\\\[6.0pt]
\eta^{\prime}=\eta_{8}\sin\theta+\eta_{0}\cos\theta,\end{array}\right.$ (34)

with the mixing angle $\theta$ taken to be the empirical value
$\theta=-23^{\circ}$.

Table 1: Model parameters associated with heavy quarks. The charm quark mass $m_{c}$ and the parameters of zero point energies $a_{cc}^{0}$, $a_{cu}^{0}$, and $a_{cs}^{0}$ are in MeV. The strengths of confinement $a_{cc}$, $a_{cu}$, and $a_{cs}$ are in MeV/fm.  $m_{c}$ | $g_{c}$ | $a_{cc}$ | $a_{cu}$ | $a_{cs}$ | $a_{cc}^{0}$ | $a_{cu}^{0}$ | $a_{cs}^{0}$
---|---|---|---|---|---|---|---
$1500$ | $0.635$ | $183.8$ | $160.9$ | $130.6$ | $-61.9$ | $-95.3$ | $-53.6$
Table 2: Masses (in MeV) of charmed mesons and baryons calculated by use of the parameters listed in Table 1. The corresponding masses from PDG Workman et al. (2022) are listed in the last column. Particles | $IJ^{P}$ | Masses | PDG values
---|---|---|---
$D^{0}$ | $\frac{1}{2}0^{-}$ | $1866.9$ | $1864.84\pm 0.05$
$D^{*+}$ | $\frac{1}{2}1^{-}$ | $2011.6$ | $2010.26\pm 0.05$
$D_{s}^{\pm}$ | $00^{-}$ | $1968.4$ | $1968.35\pm 0.07$
$D_{s}^{*\pm}$ | $01^{-}$ | $2133.3$ | $2112.2\pm 0.4$
$\eta_{c}(1S)$ | $00^{-}$ | $2975.9$ | $2983.9\pm 0.4$
$J/\psi(1S)$ | $01^{-}$ | $3096.8$ | $3096.9\pm 0.006$
$\eta_{c}(2S)$ | $00^{-}$ | $3613.4$ | $3638\pm 1$
$J/\psi(2S)$ | $01^{-}$ | $3686.1$ | $3686\pm 0.01$
$\Lambda_{c}^{+}$ | $0\frac{1}{2}^{+}$ | $2245.4$ | $2286.46\pm 0.14$
$\Sigma_{c}(2455)$ | $1\frac{1}{2}^{+}$ | $2445.4$ | $2453.97\pm 0.14$
$\Sigma_{c}(2520)$ | $1\frac{3}{2}^{+}$ | $2517.6$ | $2518.41^{+0.22}_{-0.18}$
$\Xi_{c}^{+}$ | $\frac{1}{2}\frac{1}{2}^{+}$ | $2456.0$ | $2467.71\pm 0.23$
$\Xi_{c}^{{}^{\prime}+}$ | $\frac{1}{2}\frac{1}{2}^{+}$ | $2566.7$ | $2578.2\pm 0.5$
$\Xi_{c}(2645)^{+}$ | $\frac{1}{2}\frac{3}{2}^{+}$ | $2642.6$ | $2645.10\pm 0.30$
$\Omega_{c}^{0}$ | $0\frac{1}{2}^{+}$ | $2680.8$ | $2695.2\pm 1.7$
$\Omega_{c}(2770)^{0}$ | $0\frac{3}{2}^{+}$ | $2764.3$ | $2765.9\pm 2.0$

We mention that the OBE potential between a pair of light quark and antiquark
is related to that between a pair of light quarks by a $G$-parity of the
exchanged meson Huang et al. (2004b, a); Wang et al. (2007). In the present
work, the OBE potential only exists between a pair of light antiquarks, which
is the same as that between a pair of light quarks.

The model parameters for light quarks (antiquarks) are taken from our previous
work of Ref. Huang and Wang (2018), which for the first time gave a rather
consistent description in the quark model of the masses of octet and decuplet
baryon ground states, the binding energy of deuteron, and the $NN$ scattering
phase shifts and mixing parameters for partial waves up to total angular
momentum $J=6$. These parameters are: $m_{u}=m_{d}=313$ MeV, $m_{s}=470$ MeV,
$m_{\sigma^{\prime}}=m_{\kappa}=m_{\epsilon}=980$ MeV, $m_{\sigma}=569$ MeV,
$m_{\pi}=138$ MeV, $m_{K}=495$ MeV, $m_{\eta}=549$ MeV,
$m_{\eta^{\prime}}=957$ MeV, $\Lambda=1100$ MeV, $a_{uu}=58.39$ MeV/fm,
$a_{us}=69.77$ MeV/fm, $a_{ss}=95.23$ MeV/fm, $a_{uu}^{0}=-24.53$ MeV,
$a_{us}^{0}=-22.86$ MeV, $a_{ss}^{0}=-26.10$ MeV, $g_{u(s)}=1.164$, and
$\beta=1.606$. Note that the values of some of these parameters are a little
bit different from those of Ref. Huang and Wang (2018), because the
$\delta$-function in OGE potential in Ref. Huang and Wang (2018) has now been
replaced by

$\displaystyle\delta(\bm{r}_{ij})\quad\to\quad\frac{\mu_{ij}^{3}}{\pi}\frac{e^{-\mu_{ij}^{2}r_{ij}^{2}}}{\mu
r_{ij}}$ (35)

with

$\displaystyle\mu_{ij}=\beta\frac{m_{i}m_{j}}{m_{i}+m_{j}}$ (36)

in the present work to avoid the problem of collapsing ground state caused by
the fact that the $\delta$-potential is more attractive than $1/r^{2}$ and so
overpowers the kinetic energy $p^{2}/(2m)$ for a pair of scalar quarks
(antiquarks). After this replacement, the energies of octet and decuplet
baryon ground states, the binding energy of deuteron, and the $NN$ scattering
phase shifts and mixing parameters up to total angular momentum $J=6$ are
refitted. The resulted fitting quality is almost the same as that in Ref.
Huang and Wang (2018) by use of the above-mentioned parameter values.

The other model parameters are those associated with heavy charm quarks. Apart
from the charm quark mass $m_{c}$, they are $g_{c}$, $a_{cu}$, $a_{cs}$,
$a_{cc}$, $a_{cu}^{0}$, $a_{cs}^{0}$, and $a_{cc}^{0}$ from OGE and
confinement potentials. All these parameters are determined by fitting the
masses of known charmed mesons and baryons. The fitted values of these
parameters are listed in Table 1, and the masses of charmed mesons and baryons
calculated by use of these parameters are presented in Table 2, where the
experimental values Workman et al. (2022) are also listed for comparison. One
sees that the calculated masses of charmed mesons and baryons are rather close
to their experimental values.

We mention that although the chiral SU(3) quark model offers a natural
explanation for the relatively large constituent quark masses and reasonably
reproduces the energies of individual baryons as well as the $NN$ and $YN$
($Y=\Lambda,\Sigma$) scattering data, its success remains incompletely
understood within the QCD framework.

##  III Wave functions

Table 3: Color matrix elements of $\bm{\lambda}_{i}^{c}\cdot\bm{\lambda}_{j}^{c}$. | $\bm{\lambda}_{1}^{c}\cdot\bm{\lambda}_{2}^{c}$ | $\bm{\lambda}_{3}^{c}\cdot\bm{\lambda}_{4}^{c}$ | $\bm{\lambda}_{1}^{c}\cdot\bm{\lambda}_{3}^{c}$ | $\bm{\lambda}_{2}^{c}\cdot\bm{\lambda}_{4}^{c}$ | $\bm{\lambda}_{1}^{c}\cdot\bm{\lambda}_{4}^{c}$ | $\bm{\lambda}_{2}^{c}\cdot\bm{\lambda}_{3}^{c}$
---|---|---|---|---|---|---
$\langle 6\bar{6}|\hat{O}|6\bar{6}\rangle$ | $4/3$ | $4/3$ | $-10/3$ | $-10/3$ | $-10/3$ | $-10/3$
$\langle 6\bar{6}|\hat{O}|\bar{3}3\rangle$ | $0$ | $0$ | $-2\sqrt{2}$ | $-2\sqrt{2}$ | $2\sqrt{2}$ | $2\sqrt{2}$
$\langle\bar{3}3|\hat{O}|\bar{3}3\rangle$ | $-8/3$ | $-8/3$ | $-4/3$ | $-4/3$ | $-4/3$ | $-4/3$
Table 4: Flavor matrix elements of $1$, $\sum_{a=1}^{3}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}$, $\sum_{a=4}^{7}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}$, and $\lambda_{\bar{q}}^{8}\lambda_{\bar{q}^{\prime}}^{8}$. | $[\bar{n}\bar{n}^{\prime}]$ | $\\{\bar{n}\bar{n}^{\prime}\\}$ | $[\bar{n}\bar{s}]$ | $\\{\bar{n}\bar{s}\\}$ | $\\{\bar{s}\bar{s}\\}$
---|---|---|---|---|---
$\langle 1\rangle$ | $1$ | $1$ | $1$ | $1$ | $1$
$\langle\sum_{a=1}^{3}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}\rangle$ | $-3$ | $1$ | $0$ | $0$ | $0$
$\langle\sum_{a=4}^{7}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}\rangle$ | $0$ | $0$ | $-2$ | $2$ | $0$
$\langle\lambda_{\bar{q}}^{8}\lambda_{\bar{q}^{\prime}}^{8}\rangle$ | $1/3$ | $1/3$ | $-2/3$ | $-2/3$ | $4/3$
Table 5: Spin matrix elements of $\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$. | $\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}$ | $\bm{\sigma}_{3}\cdot\bm{\sigma}_{4}$ | $\bm{\sigma}_{1}\cdot\bm{\sigma}_{3}$ | $\bm{\sigma}_{2}\cdot\bm{\sigma}_{4}$ | $\bm{\sigma}_{1}\cdot\bm{\sigma}_{4}$ | $\bm{\sigma}_{2}\cdot\bm{\sigma}_{3}$
---|---|---|---|---|---|---
$\langle\chi_{0}^{00}|\hat{O}|\chi_{0}^{00}\rangle$ | $-3$ | $-3$ | $0$ | $0$ | $0$ | $0$
$\langle\chi_{0}^{11}|\hat{O}|\chi_{0}^{11}\rangle$ | $1$ | $1$ | $-2$ | $-2$ | $-2$ | $-2$
$\langle\chi_{0}^{00}|\hat{O}|\chi_{0}^{11}\rangle$ | $0$ | $0$ | $-\sqrt{3}$ | $-\sqrt{3}$ | $\sqrt{3}$ | $\sqrt{3}$
$\langle\chi_{1}^{01}|\hat{O}|\chi_{1}^{01}\rangle$ | $-3$ | $1$ | $0$ | $0$ | $0$ | $0$
$\langle\chi_{1}^{10}|\hat{O}|\chi_{1}^{10}\rangle$ | $1$ | $-3$ | $0$ | $0$ | $0$ | $0$
$\langle\chi_{1}^{11}|\hat{O}|\chi_{1}^{11}\rangle$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$
$\langle\chi_{1}^{01}|\hat{O}|\chi_{1}^{10}\rangle$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$
$\langle\chi_{1}^{01}|\hat{O}|\chi_{1}^{11}\rangle$ | $0$ | $0$ | $-\sqrt{2}$ | $\sqrt{2}$ | $-\sqrt{2}$ | $\sqrt{2}$
$\langle\chi_{1}^{10}|\hat{O}|\chi_{1}^{11}\rangle$ | $0$ | $0$ | $\sqrt{2}$ | $-\sqrt{2}$ | $-\sqrt{2}$ | $\sqrt{2}$
$\langle\chi_{2}^{11}|\hat{O}|\chi_{2}^{11}\rangle$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
Table 6: Wave functions of the $S$\- and $P$-wave tetraquark states $cc\bar{q}\bar{q}^{\prime}$ $(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$. The superscript and subscript represent the color SU(3) representation and the spin, respectively. $J^{P}$ | Configuration
---|---
$0^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime}\\}_{1}^{3_{c}}\big{)}_{S=0}^{1_{c}}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}\\{\bar{q}\bar{q}^{\prime}\\}_{0}^{\bar{6}_{c}}\big{)}_{S=0}^{1_{c}}$
$1^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime}]_{0}^{3_{c}}\big{)}_{S=1}^{1_{c}}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{q}\bar{q}^{\prime}]_{1}^{\bar{6}_{c}}\big{)}_{S=1}^{1_{c}}$
| $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime}\\}_{1}^{3_{c}}\big{)}_{S=1}^{1_{c}}$ |
$2^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime}\\}_{1}^{3_{c}}\big{)}_{S=2}^{1_{c}}$ |
$0^{-}$ | $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{q}\bar{q}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}},\lambda\big{)}^{1_{c}}_{S=1}$
| $\big{(}\\{cc,\rho\\}_{0}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}^{6_{c}}_{1}{\\{\bar{q}\bar{q}^{\prime}\\}}^{\bar{6}_{c}}_{0}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{0}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ |
$1^{-}$ | $\big{(}\\{cc,\rho\\}_{0}^{\bar{3}_{c}}{[\bar{q}\bar{q}^{\prime}]}_{0}^{3_{c}}\big{)}^{1_{c}}_{S=0}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=0}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=0}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=2}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=2}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{0}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=0}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{q}\bar{q}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}},\lambda\big{)}^{1_{c}}_{S=1}$
| $\big{(}\\{cc,\rho\\}_{0}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}^{6_{c}}_{1}{\\{\bar{q}\bar{q}^{\prime}\\}}^{\bar{6}_{c}}_{0}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{0}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=0}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{0}^{\bar{6}_{c}},\lambda\big{)}^{1_{c}}_{S=0}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ |
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=2}$ |
$2^{-}$ | $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=2}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=2}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{q}\bar{q}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}},\lambda\big{)}^{1_{c}}_{S=1}$
| $\big{(}\\{cc,\rho\\}_{0}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}\\{cc,\rho\\}^{6_{c}}_{1}{\\{\bar{q}\bar{q}^{\prime}\\}}^{\bar{6}_{c}}_{0}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{0}^{3_{c}}\big{)}^{1_{c}}_{S=1}$ | $\big{(}{\\{cc\\}}_{0}^{6_{c}}\\{\bar{q}\bar{q}^{\prime},\rho\\}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ |
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=2}$ |
$3^{-}$ | $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}[\bar{q}\bar{q}^{\prime},\rho]_{1}^{3_{c}}\big{)}^{1_{c}}_{S=2}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{q}\bar{q}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=2}$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{q}\bar{q}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=2}$ |

In this section, we briefly present the process of constructing the wave
functions for the $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems. The total wave
function of a tetraquark state consists of four parts, i.e. the color, flavor,
spin, and space parts.

In the color space, the exact symmetry is SU(3)c. As a quark belongs to a
${\bf{3}}_{c}$ representation and an antiquark belongs to a
${\bf{\bar{3}}}_{c}$ representation, two quarks couple into either
${\bf{6}}_{c}$ or ${\bf{\bar{3}}}_{c}$ representations, i.e.
${\bf{3}}_{c}\otimes{\bf{3}}_{c}={\bf{6}}_{c}\oplus{\bf{\bar{3}}}_{c}$, and
two antiquarks couple into either ${\bf{3}}_{c}$ or ${\bf{\bar{6}}}_{c}$
representations, i.e.
${\bf{\bar{3}}}_{c}\otimes{\bf{\bar{3}}}_{c}={\bf{3}}_{c}\oplus{\bf{\bar{6}}}_{c}$.
Then the color wave function for a tetraquark state can be constructed as
either
$\big{[}\left({\bf{3}}_{c}\otimes{\bf{3}}_{c}\right)_{{\bf{6}}_{c}}\otimes\left({\bf{\bar{3}}}_{c}\otimes{\bf{\bar{3}}}_{c}\right)_{{\bf{\bar{6}}}_{c}}\big{]}_{{\bf{1}}_{c}}$
or
$\big{[}\left({\bf{3}}_{c}\otimes{\bf{3}}_{c}\right)_{{\bf{\bar{3}}}_{c}}\otimes\left({\bf{\bar{3}}}_{c}\otimes{\bf{\bar{3}}}_{c}\right)_{{\bf{3}}_{c}}\big{]}_{{\bf{1}}_{c}}$,
which can be illustrated by

$\left[\,\Big{(}~{}\yng(1)~{}\otimes~{}\yng(1)~{}\Big{)}_{~{}\scriptsize\yng(2)}~{}\otimes~{}\left(~{}\yng(1,1)~{}\otimes~{}\yng(1,1)~{}\right)_{~{}\scriptsize\yng(2,2)}\;\right]_{~{}\scriptsize\yng(2,2,2)}$

and

$\left[\,\Big{(}~{}\yng(1)~{}\otimes~{}\yng(1)~{}\Big{)}_{~{}\scriptsize\yng(1,1)}~{}\otimes~{}\left(~{}\yng(1,1)~{}\otimes~{}\yng(1,1)~{}\right)_{~{}\scriptsize\yng(2,1,1)}\;\right]_{~{}\scriptsize\yng(2,2,2)}$

respectively. In the following parts of the paper, these two types of color
wave functions are denoted as

$\displaystyle\left|6\bar{6}\right\rangle$
$\displaystyle\equiv\left[{\left(cc\right)}^{\bf{6}_{c}}{\left(\bar{q}\bar{q}^{\prime}\right)}^{{\bf\bar{6}}_{c}}\right]^{{\bf{1}}_{c}},$
(37) $\displaystyle\left|\bar{3}3\right\rangle$
$\displaystyle\equiv\left[{\left(cc\right)}^{{\bf{\bar{3}}}_{c}}{\left(\bar{q}\bar{q}^{\prime}\right)}^{{\bf{3}}_{c}}\right]^{{\bf{1}}_{c}}.$
(38)

By using the color wave functions $\left|6\bar{6}\right\rangle$ and
$\left|\bar{3}3\right\rangle$, one can evaluate the matrix elements of
$\bm{\lambda}_{i}^{c}\cdot\bm{\lambda}_{j}^{c}$ straightforwardly. The results
are presented in Table 3.

In the flavor space, one only needs to consider the symmetry of two light
antiquarks for the $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems. In this paper,
we use the curly brace $\\{~{}\\}$ and square bracket $[~{}]$ to represent the
symmetric and antisymmetric flavor wave functions of two light antiquarks,
i.e.
$\left\\{\bar{q}\bar{q}^{\prime}\right\\}=\left(\bar{q}\bar{q}^{\prime}+\bar{q}^{\prime}\bar{q}\right)/\sqrt{2}$
and
$\left[\bar{q}\bar{q}^{\prime}\right]=\left(\bar{q}\bar{q}^{\prime}-\bar{q}^{\prime}\bar{q}\right)/\sqrt{2}$.
As the chiral fields couple only to the light quarks (antiquarks), one needs
to calculate the matrix elements of the operators $1$,
$\sum_{a=1}^{3}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}$,
$\sum_{a=4}^{7}\lambda_{\bar{q}}^{a}\lambda_{\bar{q}^{\prime}}^{a}$, and
$\lambda_{\bar{q}}^{8}\lambda_{\bar{q}^{\prime}}^{8}$ in the flavor space. The
evaluated values are listed in Table 4, where $n$ and $s$ represent the non-
strange and strange quarks, respectively.

In the spin space, the wave function for a $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ system is constructed as

$\chi^{S_{12}S_{34}}_{S}\equiv{\left|{\left(cc\right)}_{S_{12}}{\left(\bar{q}\bar{q}^{\prime}\right)}_{S_{34}}\right\rangle}_{S},$
(39)

where $S_{12}$ and $S_{34}$ represent the spin of two charm quarks and two
light antiquarks, respectively, and $S$ represents the spin of the total
tetraquark system. The matrix elements of the operator
$\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ evaluated based on this spin wave
function are listed in Table 5.

In the coordinate space, we define the following Jacobi coordinates for the
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems:

$\displaystyle\bm{\xi}_{1}$ $\displaystyle=\bm{r}_{1}-\bm{r}_{2},$ (40)
$\displaystyle\bm{\xi}_{2}$ $\displaystyle=\bm{r}_{3}-\bm{r}_{4},$ (41)
$\displaystyle\bm{\xi}_{3}$
$\displaystyle=\frac{\bm{r}_{1}+\bm{r}_{2}}{2}-\frac{m_{3}\bm{r}_{3}+m_{4}\bm{r}_{4}}{m_{3}+m_{4}},$
(42)

where $\bm{r}_{1}$ and $\bm{r}_{2}$ are coordinates of two charm quarks, and
$\bm{r}_{3}$ and $\bm{r}_{4}$ are coordinates of two light antiquarks with
$m_{3}$ and $m_{4}$ being their masses, respectively. Then, the spacial wave
function of the considered tetraquark systems can be written as

$\Psi_{L}=\sum_{\alpha}A_{\alpha}\left\\{\left[\psi_{l_{1}}({\bm{\xi}}_{1})\,\psi_{l_{2}}({\bm{\xi}_{2}})\right]_{l_{12}}\psi_{l_{3}}({\bm{\xi}}_{3})\right\\}_{L},$
(43)

with

$\psi_{l_{a}}({\bm{\xi}}_{a})=\left[\frac{2^{l_{a}+2}\left(2\nu_{a}\right)^{l_{a}+\frac{3}{2}}}{\sqrt{\pi}(2l_{a}+1)!!}\right]^{1/2}{\bm{\xi}}_{a}^{l_{a}}\,e^{-\nu_{a}{\bm{\xi}}_{a}^{2}},$
(44)

where $\alpha=\\{l_{1},l_{2},l_{12},l_{3}\\}$. The expansion coefficient
$A_{\alpha}$ and the Gaussian width parameter $\nu_{a}$ will be determined by
the variational method. For $S$-wave tetraquark systems, one has
$l_{1}=l_{2}=l_{12}=l_{3}=L=0$. For $P$-wave tetraquark systems, the lowest
energy states will have one of the $l_{1}$, $l_{2}$ and $l_{3}$ being $1$
while the other two being $0$. When $l_{1}=1$ or $l_{2}=1$, the spacial wave
function will be denoted as $\rho$-mode excitation where $cc$ or
$\bar{q}\bar{q}^{\prime}$ is antisymmetric in spacial space, and when
$l_{3}=1$, the spacial wave function will be denoted as $\lambda$-mode
excitation where both $cc$ and $\bar{q}\bar{q}^{\prime}$ are symmetric in
spacial space.

With the individual wave functions in color, flavor, spin, and spacial spaces
given above, one can construct the total wave functions of the
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems
straightforwardly. All configurations considered in the present work are
listed in Table 6. There, the superscript and subscript represent the color
SU(3) representation and the spin, respectively. The curly brace $\\{~{}\\}$
and square bracket $[~{}]$ outside $cc$ and $\bar{q}\bar{q}^{\prime}$
represent the two quarks or antiquarks are symmetric or antisymmetric in
flavor space, respectively.

##  IV Results and discussion

Table 7: Masses and eigenvectors for $S$-wave $cc\bar{n}\bar{n}^{\prime}$, $cc\bar{n}\bar{s}$, and $cc\bar{s}\bar{s}$ $(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$ systems. $IJ^{P}$ | Configuration | $\langle H\rangle$ | Mass (MeV) | Eigenvector
---|---|---|---|---
$10^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{0}^{\bar{6}_{c}}\big{)}_{0}^{1_{c}}$ | $\left[\begin{array}[]{rr}4194&-81\\\ -81&4107\end{array}\right]$ | $\left[\begin{array}[]{c}4242\\\ 4059\end{array}\right]$ | $\left[\begin{array}[]{rr}-0.86&-0.51\\\ 0.51&-0.86\end{array}\right]$
|
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{1}^{3_{c}}\big{)}_{0}^{1_{c}}$
$01^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{n}\bar{n}^{\prime}]_{1}^{\bar{6}_{c}}\big{)}_{1}^{1_{c}}$ | $\left[\begin{array}[]{rr}4135&51\\\ 51&3889\end{array}\right]$ | $\left[\begin{array}[]{c}4145\\\ 3879\end{array}\right]$ | $\left[\begin{array}[]{rr}-0.98&0.20\\\ -0.20&-0.98\end{array}\right]$
|
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{n}\bar{n}^{\prime}]_{0}^{3_{c}}\big{)}_{1}^{1_{c}}$
$11^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{1}^{3_{c}}\big{)}_{1}^{1_{c}}$ | $4131$ | $4131$ | $1.00$
$12^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{1}^{3_{c}}\big{)}_{2}^{1_{c}}$ | $4176$ | $4176$ | $1.00$
$\frac{1}{2}0^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}\\{\bar{n}\bar{s}\\}_{0}^{\bar{6}_{c}}\big{)}_{0}^{1_{c}}$ | $\left[\begin{array}[]{rr}4285&-84\\\ -84&4230\end{array}\right]$ | $\left[\begin{array}[]{c}4346\\\ 4169\end{array}\right]$ | $\left[\begin{array}[]{rr}-0.81&-0.59\\\ 0.59&-0.81\end{array}\right]$
|
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{s}\\}_{1}^{3_{c}}\big{)}_{0}^{1_{c}}$
$\frac{1}{2}1^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{n}\bar{s}]_{1}^{\bar{6}_{c}}\big{)}_{1}^{1_{c}}$ | $\left[\begin{array}[]{rr}4245&52\\\ 52&4110\end{array}\right]$ | $\left[\begin{array}[]{c}4263\\\ 4092\end{array}\right]$ | $\left[\begin{array}[]{rr}-0.95&0.32\\\ -0.32&-0.95\end{array}\right]$
|
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{n}\bar{s}]_{0}^{3_{c}}\big{)}_{1}^{1_{c}}$
$\frac{1}{2}1^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{s}\\}_{1}^{3_{c}}\big{)}_{1}^{1_{c}}$ | $4255$ | $4255$ | $1.00$
$\frac{1}{2}2^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{s}\\}_{1}^{3_{c}}\big{)}_{2}^{1_{c}}$ | $4301$ | $4301$ | $1.00$
$00^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{s}\bar{s}\\}_{1}^{3_{c}}\big{)}_{0}^{1_{c}}$ | $\left[\begin{array}[]{rr}4370&-86\\\ -86&4347\end{array}\right]$ | $\left[\begin{array}[]{c}4445\\\ 4272\end{array}\right]$ | $\left[\begin{array}[]{rr}-0.75&-0.66\\\ 0.66&-0.75\end{array}\right]$
|
$\big{(}\\{cc\\}_{0}^{6_{c}}\\{\bar{s}\bar{s}\\}_{0}^{\bar{6}_{c}}\big{)}_{0}^{1_{c}}$
$01^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{s}\bar{s}\\}_{1}^{3_{c}}\big{)}_{1}^{1_{c}}$ | $4374$ | $4374$ | $1.00$
$02^{+}$ | $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{s}\bar{s}\\}_{1}^{3_{c}}\big{)}_{2}^{1_{c}}$ | $4423$ | $4423$ | $1.00$
Table 8: Individual contributions of the Hamiltonian to the $cc\bar{n}\bar{n}^{\prime}$ $(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$ systems. $IJ^{P}$ | Configuration | $\langle H\rangle$ | $\langle T\rangle$ | $\langle V^{\text{conf}}\rangle$ | $\langle V^{\text{coul}}\rangle$ | $\langle V^{\text{CM}}\rangle$ | $\langle V^{\sigma}\rangle$ | $\langle V^{\pi}\rangle$ | $\langle V^{\text{ten}}\rangle$ | $\langle V^{\text{ls}}\rangle$
---|---|---|---|---|---|---|---|---|---|---
$01^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{n}\bar{n}^{\prime}]_{1}^{\bar{6}_{c}}\big{)}_{S=1}^{1_{c}}$ | $4135$ | $713$ | $193$ | $-618$ | $190$ | $17$ | $25$ | $0$ | $0$
| $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{n}\bar{n}^{\prime}]_{0}^{3_{c}}\big{)}_{S=1}^{1_{c}}$ | $3889$ | $1015$ | $197$ | $-774$ | $1$ | $31$ | $-194$ | $0$ | $0$
$10^{+}$ | $\big{(}\\{cc\\}_{0}^{6_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{0}^{\bar{6}_{c}}\big{)}_{S=0}^{1_{c}}$ | $4194$ | $675$ | $234$ | $-602$ | $256$ | $-34$ | $39$ | $0$ | $0$
| $\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}\\{\bar{n}\bar{n}^{\prime}\\}_{1}^{3_{c}})_{S=0}^{1_{c}}$ | $4107$ | $723$ | $295$ | $-653$ | $173$ | $-40$ | $-16$ | $0$ | $0$
$00^{-}$ | $\big{(}\\{cc,\rho\\}_{1}^{6_{c}}{[\bar{n}\bar{n}^{\prime}]}_{1}^{\bar{6}_{c}}\big{)}^{1_{c}}_{S=1}$ | $4167$ | $854$ | $192$ | $-605$ | $106$ | $6$ | $28$ | $-13$ | $-27$
| $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{n}\bar{n}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ | $4150$ | $1159$ | $338$ | $-699$ | $-59$ | $17$ | $-196$ | $0$ | $-35$
$10^{-}$ | $\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{\\{\bar{n}\bar{n}^{\prime}\\}}_{1}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$ | $4393$ | $817$ | $456$ | $-581$ | $158$ | $-39$ | $-15$ | $-9$ | $-20$
| $\big{(}\\{cc,\rho\\}^{6_{c}}_{1}{\\{\bar{n}\bar{n}^{\prime}\\}}^{\bar{6}_{c}}_{0}\big{)}^{1_{c}}_{S=1}$ | $4299$ | $749$ | $287$ | $-569$ | $220$ | $-35$ | $39$ | $1$ | $-20$

Figure 1: The mass spectra of $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$ $(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ tetraquark states. Subfigures (a)-(c) correspond to the mass spectra of $cc\bar{n}\bar{n}^{\prime}$, $cc\bar{n}\bar{s}$, and $cc\bar{s}\bar{s}$ $(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$ states, respectively. For $P$-wave states, only those with lower masses are shown for the sake of simplicity. Table 9: Root mean square radius (in fm) of the $cc\bar{u}\bar{d}$ state with $IJ^{P}=01^{+}$ and mass $3879$ MeV. $IJ^{P}$ | Mass | $\sqrt{\langle r_{12}^{2}\rangle}$ | $\sqrt{\langle r_{34}^{2}\rangle}$ | $\sqrt{\langle r_{13}^{2}\rangle}$ | $\sqrt{\langle r_{24}^{2}\rangle}$ | $\sqrt{\langle r_{14}^{2}\rangle}$ | $\sqrt{\langle r_{23}^{2}\rangle}$ | $\sqrt{{\sum_{i=1}^{4}\langle({\vec{r}}_{i}-{\vec{R}_{\rm cm}})^{2}\rangle}/4}$
---|---|---|---|---|---|---|---|---
$01^{+}$ | $3879$ | $0.54$ | $0.70$ | $0.76$ | $0.76$ | $0.76$ | $0.76$ | $0.48$
Figure 2: Dependence of the mass for the lowest $cc{\bar{u}}{\bar{d}}$ state
with $IJ^{P}=01^{+}$ on parameter values. The vertical and horizontal axes
indicate the multiples of the mass and parameter values, respectively,
compared to their original values.

The reliability of the employed (anti)quark-(anti)quark interactions is
crucial for a credible understanding of the multiquark states in a quark
model. In the present work, we study the $S$\- and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ tetraquark states in a
chiral SU(3) quark model, where the Hamiltonian incorporates the kinetic
energies, the OGE potential, the phenomenological confinement potential, and
the OBE potential stemming from the couplings of light quarks and chiral
fields. The interactions between light antiquarks $\bar{q}$ and
$\bar{q}^{\prime}$ $(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ are
taken from our previous work of Ref. Huang and Wang (2018), which has been
shown to be quite successful in reproducing the energies of octet and decuplet
baryon ground states, the binding energy of deuteron, and the $NN$ scattering
phase shifts and mixing parameters up to total angular momentum $J=6$ in a
rather consistent manner. The interactions associated with charm quark,
consisting of the OGE potential and the confinement potential, are determined
by a good fit to the masses of known charmed mesons and baryons which are
listed in Table 2. We choose the Gaussian functions as the trial wave
functions in coordinate space and combine them with the color, flavor, and
spin wave functions to get the total wave functions of the $S$\- and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems that satisfy the
Pauli principle.

Equipped with the Hamiltonian and the trial wave functions, we solve the
Schrödinger equation by use of the variational method to get the masses and
the corresponding eigenvectors for the $S$\- and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states. The results for
$S$-wave $cc\bar{n}\bar{n}^{\prime}$, $cc\bar{n}\bar{s}$, and
$cc\bar{s}\bar{s}$ $(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$ systems are
listed in Table 7. One sees from this table that the lowest $S$-wave states
for $cc\bar{n}\bar{n}^{\prime}$, $cc\bar{n}\bar{s}$, and $cc\bar{s}\bar{s}$
configurations have masses of $3879$ MeV, $4092$ MeV, and $4272$ MeV,
respectively, and have quantum numbers of isospin and spin-parity
$IJ^{P}=01^{+}$, $\frac{1}{2}1^{+}$, and $00^{+}$, respectively. The effects
of channel couplings of the states with the same quantum numbers are
considerable in reducing the energies of these states. Note that there are
three $IJ^{P}=\frac{1}{2}1^{+}$ states for $cc\bar{n}\bar{s}$ as listed in
Table 6. Two of them have symmetric flavor wave functions for
$\bar{n}\bar{s}$, and they do not couple to the other state that has
antisymmetric flavor wave functions for $\bar{n}\bar{s}$.

In Table 8, we show the individual contributions of the Hamiltonian to the
$cc\bar{n}\bar{n}^{\prime}$ $(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$
systems with quantum numbers $IJ^{P}=01^{+}$, $10^{+}$, $00^{-}$, and $10^{-}$
(only the lowest two states are listed for negative parity). One sees that for
all these states, apart from the kinetic energies, the dominant contributions
are coming from the confinement potential and the color coulomb interaction of
the OGE potential, followed by the color magnetic interaction of the OGE
potential. Significant contributions are also seen from the OBE potential,
which reduces the energies of the $cc\bar{n}\bar{n}^{\prime}$ states with
$IJ^{P}=01^{+}$, $10^{+}$, and $00^{-}$ by $163$ MeV, $56$ MeV, and $179$ MeV,
respectively. Note that the contributions of $\pi$ exchange are proportional
to the matrix elements of the operator
$\langle\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\rangle$
where $\tau^{a}=\lambda^{a}$ ($a=1,2,3$). For the states
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{n}\bar{n}^{\prime}]_{0}^{3_{c}}\big{)}_{S=1}^{1_{c}}$
with $IJ^{P}=01^{+}$ and
$\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{n}\bar{n}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$
with $IJ^{P}=00^{-}$, one has
$\langle\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}\,\bm{\tau}_{i}\cdot\bm{\tau}_{j}\rangle=9$,
much bigger than $1$ or $-3$ in other states, as here both isospin and spin of
$\bar{n}\bar{n}^{\prime}$ are zero, which results in the maximum attractions
of $\pi$ exchange, reducing the energies of these two states by $194$ MeV and
$196$ MeV, respectively. The kinetic energies of these two states are observed
to be bigger than the other states. This is because that the strong
attractions offered by $\pi$ exchange in these two states compress the spacial
sizes of the light quarks in these two states while the kinetic energies are
in inverse proportion to the distances of quarks. The contributions from the
tensor force and spin-orbit force, marked as $\langle V^{\text{ten}}\rangle$
and $\langle V^{\text{ls}}\rangle$ in the last two columns, are seen to be
rather small. This means that the mixing between states with the same quantum
numbers $IJ^{P}$ but different excitation modes in orbit space (excitations of
the relative motion between $cc$, the relative motion between
$\bar{q}\bar{q}^{\prime}$, and the relative motion between $cc$ and
$\bar{q}\bar{q}^{\prime}$) are not considerable and the physical states may be
approximated to have pure excitation modes.

In Fig. 1, we present the calculated mass spectra of $S$-wave and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states. For $P$-wave
states, only those with lower masses are shown for the sake of simplicity. The
meson-meson thresholds of the possible fall-apart decay channels are marked
with dashed lines. One sees from Fig. 1 that the $cc\bar{n}\bar{n}^{\prime}$
$(\bar{n},\bar{n}^{\prime}=\bar{u},\bar{d})$ state with $IJ^{P}=01^{+}$ just
locates at $3879$ MeV, very close to the calculated $DD^{*}$ threshold. This
state cannot decay to $DD$, and thus is expected to have a rather small decay
width. Experimentally, a narrow state $T_{cc}^{+}(3875)$ with quantum numbers
$IJ^{P}=01^{+}$ and quark constituent $cc\bar{u}\bar{d}$ has been reported by
the LHCb Collaboration Aaij et al. (2022a, b). The state with mass $3879$ MeV
predicted in our model has the same quantum numbers and quark constituent with
the experimentally observed $T_{cc}^{+}(3875)$. In Table 9, we present the
root mean square radius of this state calculated in our chiral SU(3) quark
model. One sees that the root mean square radius of this state,
$\sqrt{{\sum_{i=1}^{4}\langle({\vec{r}}_{i}-{\vec{R}_{\rm
cm}})^{2}\rangle}/4}$, is only $0.48$ fm, and moreover, the distances of any
pairs of quarks (antiquarks) are less than or equal to $0.76$ fm. This
indicates that this state has a rather compact structure. All these
demonstrates that the experimentally observed $T_{cc}^{+}(3875)$ state can be
accommodated as a stable and compact $cc\bar{u}\bar{d}$ tetraquark sate in the
chiral SU(3) quark model. This state has $4\%$ of the
$\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{u}\bar{d}]_{1}^{\bar{6}_{c}}\big{)}_{1}^{1_{c}}$
configuration and $96\%$ of the
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{u}\bar{d}]_{0}^{3_{c}}\big{)}_{1}^{1_{c}}$
configuration, as can be seen from the eigenvectors listed in Table 7.

One also sees from Figs. 1-1 that all the other $S$\- and $P$-wave
$cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states have masses about
one hundred to few hundreds MeV higher than the corresponding meson-meson
thresholds. For the state
$\big{(}{\\{cc\\}}_{1}^{\bar{3}_{c}}{[\bar{n}\bar{n}^{\prime}]}_{0}^{3_{c}},\lambda\big{)}^{1_{c}}_{S=1}$
with $IJ^{P}=00^{-}$, although the $\pi$ exchange offers strong attractions
with similar strength to the lowest state
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{n}\bar{n}^{\prime}]_{0}^{3_{c}}\big{)}_{S=1}^{1_{c}}$
with $IJ^{P}=01^{+}$, the color-coulomb force offers much less attraction and
the confinement potential offers much bigger repulsion. Finally this state has
a mass more than two hundred MeV higher than the lowest state and the
threshold of $DD^{\ast}$. As these states locate much higher than the
corresponding meson-meson thresholds, they are expected to decay easily into
two mesons via fall-apart mechanism. Thus, these states are not suggested to
be candidates of stable and compact tetraquark states in the chiral SU(3)
quark model.

In order to see to which extent do our calculated masses of the
$cc{\bar{q}}{\bar{q}}^{\prime}$
(${\bar{q}},{\bar{q}}^{\prime}={\bar{u}},{\bar{d}}$) tetraquark states depend
on the values of the model parameters, in Fig. 2, we illustrate the dependence
of the mass for the lowest $cc{\bar{u}}{\bar{d}}$ state with $IJ^{P}=01^{+}$
on the parameter values. Here the vertical and horizontal axes indicate the
multiples of the mass and parameter values, respectively, compared to their
original values. It is observed that when the value of each of the parameter
of $g_{c}$, $a_{cu}$, $a_{cc}$, $a^{0}_{cu}$, and $a^{0}_{cc}$ varies by
$10\%$, the mass of the lowest $cc{\bar{u}}{\bar{d}}$ state with
$IJ^{P}=01^{+}$ changes less than $2\%$. This demonstrates that our calculated
results are relatively stable against the values of these parameters.
Nevertheless, when the charm quark mass, $m_{c}$, varies by $10\%$, the mass
of the lowest $cc{\bar{u}}{\bar{d}}$ state with $IJ^{P}=01^{+}$ fluctuates by
about $7\%$. This can be understood if we notice that $2m_{c}$ makes up the
bulk of the mass of the $cc{\bar{u}}{\bar{d}}$ states. In practical
calculation, one needs to refit the other parameters when a different value of
$m_{c}$ is chosen. By doing so, the resulting mass of the
$cc{\bar{u}}{\bar{d}}$ state with $IJ^{P}=01^{+}$ keeps nearly unchanged as
depicted in Fig. 2 with a dashed line.

##  V Summary

In this work, we systematically explore the mass spectra of the $S$\- and
$P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ systems in a chiral SU(3)
quark model. The Hamiltonian consists of the kinetic energy, the confinement
potential, the OGE potential, and the OBE potential stemming from the coupling
of light quarks (antiquarks) and chiral fields. The interactions between the
light antiquarks $\bar{q}$ and $\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ are taken from our
previous work of Ref. Huang and Wang (2018), that reproduced the energies of
octet and decuplet baryon ground states, the binding energy of deuteron, and
the $NN$ scattering phase shifts and mixing parameters for total angular
momentum up to $J=6$ quite well in a rather consistent manner. The
interactions associated with charm quarks are fixed by fitting the masses of
known charmed baryons and mesons. The energies and eigenvectors of the $S$\-
and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states are obtained by
solving the Schrödinger equation via the variational method.

The results show that the lowest $S$-wave $cc\bar{u}\bar{d}$ state with
quantum numbers $IJ^{P}=01^{+}$ has a mass $3879$ MeV, approximately at the
calculated threshold of $DD^{\ast}$. This state has $4\%$ of the
$\big{(}\\{cc\\}_{0}^{6_{c}}[\bar{u}\bar{d}]_{1}^{\bar{6}_{c}}\big{)}_{1}^{1_{c}}$
configuration and $96\%$ of the
$\big{(}\\{cc\\}_{1}^{\bar{3}_{c}}[\bar{u}\bar{d}]_{0}^{3_{c}}\big{)}_{1}^{1_{c}}$
configuration. It cannot decay to the $DD$ channel due to its quantum numbers,
and thus is expected to have a rather small decay width. All the mass, width,
and quantum numbers of this state are consistent with those of the narrow
state $T_{cc}^{+}(3875)$ reported by the LHCb Collaboration Aaij et al.
(2022a, b). Moreover, the calculated root mean square radius of this state is
$\sqrt{{\sum_{i=1}^{4}\langle({\vec{r}}_{i}-{\vec{R}_{\rm
cm}})^{2}\rangle}/4}=0.48$ fm, and the distances of any pair of quarks
(antiquarks) are less than or equal to $0.76$ fm. All these demonstrate that
the experimentally observed $T_{cc}^{+}(3875)$ state can be accommodated as a
compact $cc\bar{u}\bar{d}$ tetraquark state in the chiral SU(3) quark model.

The energies of all other $S$\- and $P$-wave $cc\bar{q}\bar{q}^{\prime}$
$(\bar{q},\bar{q}^{\prime}=\bar{u},\bar{d},\bar{s})$ states are about one
hundred to several hundreds MeV higher than the thresholds of the
corresponding meson-meson channels that can be decayed into via fall-apart
mechanism. Thus, they are expected to have large decay widths and not
suggested as candidates of compact tetraquark states in the chiral SU(3) quark
model.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Grants No. 12175240 and No. 11635009, and the Fundamental Research
Funds for the Central Universities.

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