# On the Robustness of Language Models for Tabular Question Answering

Kushal Raj Bhandari*$\diamondsuit$, Sixue Xing$\diamondsuit$, Soham
Dan$\clubsuit$, Jianxi Gao$\diamondsuit$,
$\diamondsuit$Rensselaer Polytechnic Institute, $\clubsuit$IBM Research,
Correspondence<EMAIL_ADDRESS>
###### Abstract

Large Language Models (LLMs), originally shown to ace various text
comprehension tasks have also remarkably been shown to tackle table
comprehension tasks without specific training. While previous research has
explored LLM capabilities with tabular dataset tasks, our study assesses the
influence of in-context learning, model scale, instruction tuning, and domain
biases on Tabular Question Answering (TQA). We evaluate the robustness of LLMs
on Wikipedia-based WTQ and financial report-based TAT-QA TQA datasets,
focusing on their ability to robustly interpret tabular data under various
augmentations and perturbations. Our findings indicate that instructions
significantly enhance performance, with recent models like Llama3 exhibiting
greater robustness over earlier versions. However, data contamination and
practical reliability issues persist, especially with WTQ. We highlight the
need for improved methodologies, including structure-aware self-attention
mechanisms and better handling of domain-specific tabular data, to develop
more reliable LLMs for table comprehension.

On the Robustness of Language Models for Tabular Question Answering

Kushal Raj Bhandari*$\diamondsuit$, Sixue Xing$\diamondsuit$, Soham
Dan$\clubsuit$, Jianxi Gao$\diamondsuit$, $\diamondsuit$Rensselaer Polytechnic
Institute, $\clubsuit$IBM Research, Correspondence<EMAIL_ADDRESS>

Figure 1: Example of different possible table augmentation methods where each
row operation represents original table augmentation, transpose-based table
augmentation, and value-based perturbation. The value-based perturbation only
affects the cell content associated with the answer value of the given
question.

##  1 Introduction

LLMs, primarily trained on unstructured data, have performed fairly well with
structured data like tabular datasets. Tasks that rely on tabular structures
require the ability to process and understand data with structural awareness,
even when the input is provided in a sequential linear format. Exploring such
capability—particularly how LLMs apprehend and interpret structured
information—can be effectively analyzed by studying robustness measures in
table comprehension. This approach offers insights into the models’ ability to
navigate and interpret the inherent relationships within structured datasets
Borisov et al. (2023).

Tabular Question Answering (TQA) is a popular task on tables, which involves
answering a query based on the given structured data Herzig et al. (2020);
Fang et al. (2024). TQA task highlights the emergent properties of LLMs,
demonstrating their capability to comprehend structured data through in-
context learning, instruction adherence, and multi-step reasoning Fang et al.
(2024); Wei et al. (2022); Wang et al. (2024); Jiang et al. (2023); Dong et
al. (2024); Liu et al. (2024). This capability can be quantitatively evaluated
by measuring the robustness of the model’s performance on tabular datasets.
Wang et al. (2022) recently showed the vulnerability of LLMs to structural
augmentation of the input table and introduced a novel methodology to leverage
structure-aware self-attention to more effectively manage table
transformations, thereby improving the performance of natural language
generation from tables. Similarly, Zhao et al. (2023) identified the
vulnerabilities of recent TQA LLMs, when faced with human-crafted adversarial
perturbations. Liu et al. (2023) discusses inherent limitations within LLMs
for tabular data interpretation, arguing that these models struggle with
structural perturbations and advocate for an integrated approach that combines
textual and symbolic reasoning. Chen (2023) find the performance of few-shot
reasoning with LLMs for TQA show strong complex reasoning abilities over table
structures without table-specific pre-training.

Although these studies showcase a critical need for enhanced methodologies in
handling tabular data across various applications, they have yet to fully
explore how the performance of robustness with the inherent ability of LLMs
varies with different factors. Notably, there has been little comparison of
robustness as a function of (1) in-context learning, (2) the scale and
instruction-tuning of LLMs, and (3) the biases inherent within existing
evaluation benchmarks for robust table-to-text generation. These aspects are
crucial, as the scale of LLMs can influence their ability to generalize from
data, while instruction-tuning enhances in-context learning ability, affecting
how well they adapt to table comprehension without explicit re-training.
Additionally, biases in training data can skew the evaluation of their
comprehension ability. Hence, understanding these dimensions is essential for
developing more reliable and fair LLMs in processing tabular data. To evaluate
these, Wikipedia-based datasets are the most popular benchmark datasets for
TQA. We experiment with WTQ dataset, a Wikipedia-based tabular question-
answering dataset Pasupat and Liang (2015) and TAT-QA Zhu et al. (2021), a
tabular question-answering dataset derived from financial reports. A general
and specialized domain ensures reliable benchmarking of the robustness and
generalizability of existing LLMs.

##  2 Various Perturbation Categories

Each perturbation is designed to manipulate table structure or content while
preserving the inherent relational meaning of the table, to measure robustness
to table comprehension.

###  2.1 Structural Perturbation (SP)

SP involves rearranging the columns and rows of the table to generate new
examples. This ensures flexibility in understanding tabular data without
distorting the semantics of the table. SP involves column swap, row swap,
transpose, transpose column swap, and transpose row swap.

###  2.2 Value Perturbation (VP)

VP focuses on modifying the actual data values within tables, ensuring that
the model is faithful to the table. We explore two types of VP:
Data Type Preserving Perturbation (DVP) DVP involves altering the answer to
the question and, respectively, the cell values within the table while
maintaining their original data types. For instance, in Fig 7, given a
question, ‘‘What was the first venue for the Asian games?’’, we modify the
correct answer, ‘‘Bangkok, Thailand", to ‘‘Beijing’’. We use such
counterfactual entities to test faithfulness of the LLM to the tabular data.
111 We filter out the subset of data points where the table does not contain
the answer, e.g., ‘‘How many people stayed at least 3 years in office?’’. We
use an automated LLM-based counterfactual answer generation by prompting
GPT-3.5, which ensures type-correctness of the altered answer. 222The full
prompts are present in the Appendix.

Random Value Perturbation (RVP) RVP (an example is shown in Fig. 8) relaxes
DVP where instead of a counterfactual entity we have a fixed string, e.g.,
‘‘r@nD0m v@1u3’’.

####  2.2.1 Null Value Perturbation (NVP)

NVP removes the correct answer from the table completely. Evaluating the
performance on NVP highlights the influence of Wikipedia content on solving
the WTQ table question-answering task.

####  2.2.2 No Table (NT)

To understand the extent of bias in WTQ, we evaluate LLMs on the no-table
baseline. By analyzing the performance of LLMs in the absence of the table, we
can better understand the extent to which these models depend on the tabular
data.

##  3 Results

Figure 2: The average EM score for the different few shot comparisons for
various perturbations

###  3.1 Effects of ICL examples on TQA

Does instruction prompting assists LLM for better table comprehension for
question answering task?

Fig. 10 compares the performance of various LLMs under different few-shot
settings and perturbations, measured by exact match (EM) scores. The first
plot illustrates the performances in 0-, 1-, and 2-shot scenarios. Notably,
there’s an increasing trend in EM with the number of shots, demonstrating that
additional examples improve table comprehension abilities. Thus, we
demonstrate complementary results to LLMs on unstructured tasks Wei et al.
(2022) in the context of TQA. Fig. 2 focuses on the impact of various
perturbations on model performance. Here the models display a drop in EM
across different manipulation operations which indicates a sensitivity to
changes in the data structure, underscoring a potential vulnerability in model
robustness against input invariances and equivariances. The consistency in
performance degradation across operations suggests that while models can
handle the original task, their structural awareness ability still requires
substantial enhancement for broader table comprehension.

###  3.2 Effects of the Model Type on TQA

Does newer models have better TQA abilities?

Fig 3 shows that Llama3 models generally outperform the Llama2 and Mistral
models across different configurations, indicating that newer architectures
like Llama3 are more effective at table reasoning tasks. The 70b versions of
these models generally perform better than their 7b variants, indicating that
larger model sizes enhance reasoning capabilities. Overall, the advancements
in model architecture and increased model size significantly contribute to
better TQA abilities.

Figure 3: The average EM for different models across the WTQ dataset. The
models are categorized by their types (Llama2, Llama3, Mistral) and separated
by their sizes (7b, 70b, etc.)

Does instruction tuning help TQA?

The heat maps in Fig. 4 illustrate the performance of various LLMs,
demonstrating that models fine-tuned for instructions or conversation exhibit
improved performance. For instance, the Llama3-70b-Instruct model
significantly outperforms its original version across all table
transformations, indicating that instruction-based fine-tuning enhances the
model’s ability to handle complex reasoning tasks. Similarly, conversation-
focused fine-tuning also leads to better scores, though with a less pronounced
improvement compared to instruction-focused tuning. This suggests that fine-
tuning models on specific tasks like following instructions or conversing
effectively enhances their capability to interpret and manipulate tabular
data, making such approaches valuable for improving performance in structured
data tasks.

How does model size affect TQA?

Larger models (e.g., Llama3-70b, Mixtral-8x7b) generally show higher
performance than smaller models (e.g., Llama2-7b, Mistral-7b), as seen in both
the bar plot (Fig 3) and heat maps (Fig 4). For instance, Llama3-70b and
Mixtral-8x7b have higher EM scores than Llama2-7b and Mistral-7b. This
indicates that model size contributes significantly to TQA performance.

Figure 4: The average EM scores of the original models and fine-tuned for
instruction or conversation models across various table augmentation
techniques for WTQ dataset

How do performances vary with perturbations?

Table 1 shows the performance of small and large LMs when subjected to various
types of value-based perturbation on WTQ. Both model sizes experience a
decline in exact match scores across different operations compared to the
original setup. The performance with RVP results in a significant performance
drop, more so than DVP. This suggests a sharp decrease in the model’s ability
to process and comprehend tables when the insertion of arbitrary, non-
contextual values compromises the table comprehension ability of LLMs.
Conversely, the DVP result indicates that while the model struggles with
content that deviates from the original data structure, maintaining data type
consistency offers some resilience against total comprehension failure. The
observed discrepancies in performance, particularly the deviation from the
original model’s performance in scenarios like DVP and RVP, underline a
fundamental challenge: these models do not consistently apply their tabular
comprehension capabilities when faced with manipulated table values.

VP Operation | Large Model | Small Model
---|---|---
Original | 0.35 | 0.28
NT | 0.06 | 0.02
DVP | 0.20 | 0.16
RVP | 0.13 | 0.11
NVP | 0.06 | 0.02
Table 1: The average EM of small and large LLMs on Value Perturbations(VP).
The classification of large and small models is defined in Table 3

###  3.3 Effects of Domain Specificity on TQA

How biased are LLMs towards Wiki-tables?

The Table 1 outlines the reliance of small and large language models on
structured tabular data, particularly evident in the NT and NVP operations.
When no table is provided, both models show a notable performance decline,
emphasizing their dependence on tabular data to generate correct answers.
Interestingly, the models still manage to answer about $\approx$5% of queries
correctly, indicating a potential bias towards Wiki-data. In the NVP scenario,
where table values relevant to queries are nullified, there is also a
significant drop in performance, yet less severe compared to the complete
absence of a table. This suggests that the presence of the original structured
format without the relevant data provides enough contextual cues for the
models to answer, highlighting model biases.

How do LLMs perform on (TAT-QA), which is a specialized dataset that requires
stronger reasoning abilities?

Both large and small LLMs (Fig 5) exhibit moderate performance across various
table augmentations on the TAT-QA dataset. Notably, larger models (e.g.,
Llama3-70b, Mixtral-8x7b) consistently outperform smaller models (e.g.,
Llama2-7b, Mistral-7b), underscoring their superior TQA abilities. However,
the overall scores still denote significant challenges inherent in the
dataset. Fine-tuned models, such as Llama3-70b-Instruct, demonstrate enhanced
effectiveness, reflecting the advantages of model-specific optimization for
harder tabular comprehension tasks. The comparison between the TAT-QA and WTQ
datasets highlights a critical insight: while models achieve higher accuracy
on WTQ, their moderate EM scores on TAT-QA indicates lack of robustness in
TQA. Particularly the sharp contrast in the NT performance, where TAT-QA
models have approximately 0% accuracy, a distinguishable contrast from the WTQ
performance. This suggests that performance metrics on WTQ might be inflated
due to biases favoring familiarity with Wikipedia’s structured data, whereas
TAT-QA, a more specialized rarer domain is more challenging. This highlights
the need for better benchmark datasets to accurately assess LLMs’ robust table
comprehension abilities.

Figure 5: The average EM scores of the original models and fine-tuned for
instruction or conversation models across various table augmentation
techniques for TAT-QA dataset

##  4 Conclusion

In conclusion, our study explores the table comprehension abilities of LLMs
through the lens of various invariant and equivariant perturbations. We find
that instruction tuning and ICL examples significantly enhances table
comprehension ability of LLMs. Newer models, such as Llama3, exhibit robust
performance compared to their predecessors. Additionally, we uncover domain
biases for evaluating tabular data using Wikipedia-based datasets. Despite the
decent performance of LLMs on TQA, they still fall short of the reliability
required for several practical applications.

##  5 Limitation

Although we provide extensive evaluation of LLMs on WTQ and TAT-QA datasets,
it is possible to including a broader range of datasets for a more
comprehensive comparison that would highlight the generalizability of our
method for both domain specific dataset and Wikipedia-based dataset. While we
anticipate that similar performance could be achieved with other tasks, such
as table summarization, future work should include extensive analysis across
various tasks and datasets to validate the assumption. Moreover, our study did
not involve any structural aware or fine-tuned models for tabular dataset. It
is plausible that fine-tuning and structurally enhanced models could
significantly impact the performance of different models. Additionally, our
evaluation relied on exact match accuracy for assessing the text generation
model’s performance. This metric, while useful, limits the scope of evaluation
for question answering task. Future studies should employ more nuanced
evaluation metrics to better assess the robustness of the models in TQA tasks.

## References

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  * Pasupat and Liang [2015] Panupong Pasupat and Percy Liang. 2015.  Compositional Semantic Parsing on Semi-Structured Tables.  In _Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 1470–1480. Association for Computational Linguistics.
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##  Appendix A Appendix

###  A.1 Example Prompt

Example of the prompts:

Figure 6: Example of a prompt with answer for WTQ dataset without Few Shot
Prompt Figure 7: Example of a prompt with answer for WTQ dataset for Data Type
Preserving Perturbation. In comparison to Fig 6, we replace correct
answer(Bankok, Thailand) with a fake answer(Beijing). Figure 8: Example of a
prompt with answer for WTQ dataset for Random Value Perturbation. Here, we
replace correct answer(Bankok, Thailand) with an abstract random value (r@nD0m
v@1u3). Figure 9: Example of a prompt with answer for WTQ dataset for Random
Value Perturbation. We remove the correct answer(Bankok, Thailand).

##  Appendix B Evaluation Dataset Size

We select two different datasets for comparison, WTQ and TAT-QA dataset, with
different number of evaluation dataset as described in Table 2. For fair
comparison we limit the number of cell elements($<150$) within the table for
both datasets. Similarly, for Value Perturbation, some queries relate to over
all structure of the table, hence we filter only those tables that contains
the answer value for the given query.

Operation | Number of Pairs
---|---
WTQ dataset
Row Swap | 204
Column Swap | 204
Transpose | 204
Transpose Row Swap | 204
Transpose Column Swap | 204
Data Type Preserving | 141
Random Value | 141
Null Value | 141
No Table | 204
TAT-QA dataset
Row Swap | 1668
Column Swap | 1668
Transpose | 1668
Transpose Row Swap | 1668
Transpose Column Swap | 1668
No Table | 1668
Table 2: Size of the Evaluation datasize

##  Appendix C Models

We selected recent open source models that have been extensively studied.
Table 3 lists all the models we considered with their parameter size and their
date of release.

Model | Size | Date Released
---|---|---
Small Model
Llama-2-7b-hf | 6.74B | July 2023
Llama-2-7b-chat-hf | 6.74B | July 2023
Mistral-7B-v0.1 | 7.24B | Sept 2023
Mistral-7B-Instruct-v0.1 | 7.24B | Sept 2023
Meta-Llama-3-8B | 8.03B | April 2024
Meta-Llama-3-8B-Instruct | 8.03B | April 2024
Large Model
Llama-2-70b-hf | 69B | July 2023
Llama-2-70b-chat-hf | 69B | July 2023
Mixtral-8x7B-v0.1 | 46.7B | Dec 2023
Mixtral-8x7B-Instruct-v0.1 | 46.7B | Dec 2023
Meta-Llama-3-70B | 70.6B | April 2024
Meta-Llama-3-70B-Instruct | 70.6B | April 2024
Table 3: All the models with their parameter size and their date released.
Figure 10: The average EM score for the different few shot comparisons for
models on the WTQ dataset


††thanks: Author to whom correspondence should be addressed:
<EMAIL_ADDRESS>
# Integrating planar circuits with superconducting 3D microwave cavities using
tunable low-loss couplers

Ziyi Zhao  JILA, National Institute of Standards and Technology and the
University of Colorado, Boulder, CO, USA.  Department of Physics, University
of Colorado, Boulder, CO, USA.  Eva Gurra  JILA, National Institute of
Standards and Technology and the University of Colorado, Boulder, CO, USA.
Department of Physics, University of Colorado, Boulder, CO, USA.  Eric I.
Rosenthal  E. L. Ginzton Laboratory, Stanford University, Stanford, California
94305, USA  Leila R. Vale  National Institute of Standards and Technology,
Boulder, Colorado 80305, USA  Gene C. Hilton  National Institute of Standards
and Technology, Boulder, Colorado 80305, USA  K. W. Lehnert  JILA, National
Institute of Standards and Technology and the University of Colorado, Boulder,
CO, USA.  Department of Physics, University of Colorado, Boulder, CO, USA.
National Institute of Standards and Technology, Boulder, Colorado 80305, USA

###### Abstract

We design and test a low-loss interface between superconducting 3-dimensional
microwave cavities and 2-dimensional circuits, where the coupling rate is
highly tunable. This interface seamlessly integrates a loop antenna and a
Josephson junction-based coupling element. We demonstrate that the loss added
by connecting this interface to the cavity is 1.28 kHz, corresponding to an
inverse quality factor of $1/(4.5\times 10^{6})$. Furthermore, we show that
the cavity’s external coupling rate to a 50 $\Omega$ transmission line can be
tuned from negligibly small to over 3 orders of magnitude larger than its
internal loss rate in a characteristic time of 3.2 ns. This switching speed
does not impose additional limits on the coupling rate because it is much
faster than the coupling rate. Moreover, the coupler can be controlled by low
frequency signals to avoid interference with microwave signals near the cavity
or qubit frequencies. Finally, the coupling element introduces a 0.04
Hz/photon self-Kerr nonlinearity to the cavity, remaining linear in high
photon number operations.

††preprint: AIP/123-QED

To scale up and perform more demanding quantum information processing tasks,
modular quantum networks built upon superconducting circuits have emerged as a
promising approach Brecht  _et al._ (2016). This architecture distributes
entanglement over spatially separated modules to divide the scaling challenge
into independent and more manageable parts Chou  _et al._ (2018); Zhou  _et
al._ (2021); Magnard  _et al._ (2020); Niu  _et al._ (2023); Kannan  _et al._
(2023). In order for these modules to store quantum states in high quality
factor 3-dimensional microwave cavities, an ongoing challenge is to implement
rapid, reconfigurable, and low-loss swap interactions among the cavities Wang
_et al._ (2016); Reagor  _et al._ (2016); Chakram  _et al._ (2021). Because it
is hard to apply low frequency control signals to the coupling elements into
3D cavities without spoiling their quality factor, one solution uses
microwave-actuated nonlinear elements and coaxial cable modes to couple 3D
cavities Pfaff  _et al._ (2017); Burkhart  _et al._ (2021); Axline  _et al._
(2018); Spring  _et al._ (2022). In contrast, by routing the microwave signal
out of the superconducting enclosure and onto a planar circuit, where the mode
is confined between the circuit elements on the chip and the flat ground plane
beneath them, it is possible to use a much wider class of circuit elements,
including those with multiple wiring layers, magnetic flux bias, and low
frequency control. Furthermore, the characteristics of the planar circuit
elements in such devices can be accurately simulated using the well developed
planar method of moment solvers. It is then desirable to incorporate the
design flexibility and large scale manufacturability of the planar circuits
with high quality factor 3-dimensional microwave cavities with an integrated
interface.

Here, we demonstrate such an interface that seamlessly integrates a loop
antenna and a tunable coupling element on a planar circuit and connect it to a
coaxial quarter wave cavity. In our design, we utilize a symmetric coupling
element connected to a superconducting loop, which is an antenna coupled to
the magnetic field of the cavity Chapman  _et al._ (2016). The integrated
nature of this interface allows us to precisely predict all the microwave
modes in the system and eliminate seam loss in the microwave signal lines
Brecht  _et al._ (2015). To this end, our device can tune the external
coupling rate of the cavity by a factor of 1130 compared to its internal
linewidth, while introducing a loss of 1.28 kHz. This operation can be
performed with a characteristic time of $3.2$ ns and remains linear at high
intracavity photon number Campagne-Ibarcq  _et al._ (2020).

Figure 1: Device description and image a) A superconducting loop ($3.5\
$mm$\times 1.0\ $mm) (yellow) on a Si substrate (turquoise) is inductively
coupled to a 3D microwave cavity made of high purity (5N) aluminum
($\omega_{\text{cav}}/2\pi$ = 5.772 GHz) (white), which also acts as the
ground plane for the microstrip lines on the chip (see supplementary
material). The magnetic field is depicted as the black contour lines with
arrows surrounding the post, where their distance to the post signifies their
amplitude. b) An electrical schematic of the system shows that the cavity is
connected to the coupling element through a superconducting loop (yellow) and
capacitors (orange). The coupling element is a Wheatstone bridge, consisting
of SQUIDs arrays. The dark (light) green SQUID arrays experience the same flux
bias and have the same inductance, but the flux bias and inductance are
different between dark and light green SQUID arrays. The uniform (pink) and
gradiometric (blue) bias line are used to control the sum and difference flux
bias. The output is converted to single-ended mode by a balun (purple). c) A
false-color image of the chip, fabricated with the NIST niobium trilayer
process Mates  _et al._ (2008), shows the superconducting loop and the
coplanar strip (yellow) connecting it to the inductive Wheatstone bridge,
gradiometric bias line with filter (blue), uniform bias line with filter
(pink), balun (purple), SQUID arrays (green), capacitors (orange), and on-chip
ground (light grey). An on-chip ground is only used for the control lines or
for the signal path after it has left the interface via the output balun. The
insets show an array of SQUIDs, and a zoomed in picture of 2 SQUIDs.

The symmetry of the coupling element, a Tunable Inductor Bridge (TIB),
provides a low residual coupling in its off-mode (balanced) configuration, and
consequently, the cavity off-mode linewidth is dominated by its internal loss
rate Chapman  _et al._ (2016); Rosenthal  _et al._ (2021); Kerckhoff  _et al._
(2015). This symmetry is preserved throughout the signal path, which is
composed of the TIB, a coplanar strip transmission line, and the loop antenna,
a superconducting loop that is inserted into the shorted end of a quarter-wave
coaxial resonator (Fig. 1). The oscillating magnetic field of the cavity’s
resonant mode launches a differential mode to the coplanar strip transmission
line (Fig. 1a).

In addition to high on/off ratio, the integrated nature of this interface
makes it low loss; it eliminates places where the current flows between
disjoint conductors (seams). The propagating mode passing through the
interface is a conductor-backed coplanar strip mode, where the signal flows on
the metal traces on the chip, and the conductor backing is machined from the
same piece of aluminum as the cavity. In the signal path, the loop antenna and
the coupling element are already on the same chip and require no on-chip
ground reference (Fig. 1c). Without extra microwave components, for instance,
microwave connectors, cables, printed circuit boards or wirebonds to connect
the antenna and the coupling elements, the electrical length between them is
minimized and the parasitic microwave modes can be predicted during the design
process.

The tunability of this interface is achieved by controlling the TIB, which is
an inductive Wheatstone bridge Bergeal  _et al._ (2010) using Superconducting
Quantum Interference Device (SQUID) arrays that act as flux tunable inductors
whose non-linearity is suppressed by increasing the number of SQUIDs in the
array (Fig. 1b). The on/off mode of the coupler is controlled by the currents
in two bias lines, one gradiometric and one uniform (see supplementary
material), and the net flux they induce in the SQUID loop is shown in Fig. 1b.
This arrangement causes two inductors on opposite sides of the bridge to
experience the same flux and therefore have the same inductance, but
neighboring inductors have different inductance, as indicated by the different
shading of the SQUID symbols. At zero gradiometric bias, all inductances are
equal, the bridge is balanced, and the coupler is off. Because the uniform
bias only needs to be set once in the experiment, in the following
characterization, only the gradiometric bias $\Phi$ is tuned. The external
coupling rate can be tuned continuously, and this can be used in pulse shaping
schemes, but here we only characterize the “digital” behavior of our device,
where the coupling element is switched between the on and off states. Because
the cavity resonance frequency tunes with the bias, operating digitally allows
us to work with a simplified picture where the spectrum of the outgoing
microwave field from the cavity is always centered at the cavity frequency
when the switch is on.

Figure 2: Coupling rate measurements a) Total linewidth $\kappa$ (blue) and
minimal reflection amplitude $|\Gamma_{\text{min}}|$ (solid orange) are
extracted as a function of bias near the critical coupling point (dashed
orange). b) Representative samples of the time traces are shown with different
decay rates. Each trace is normalized by the maximum of the green trace
$V_{0}=\sqrt{E_{\text{stored}}~{}\kappa_{g}~{}Z_{0}}$, where $\kappa_{g}$ is
the total linewidth at the green trace bias and $Z_{0}$ is the characteristic
impedance of the transmission line connected to the output port. The inset is
the time sequence of excitations applied to the cavity and control signals to
the TIB. The different bias values correspond to the time traces with the same
colors. c) The total decay rate (blue) and measured energies
($E_{\text{meas}}\propto\int dtV^{2}$, solid orange) are extracted from time
domain measurements. $E_{\text{stored}}$ is the maximum of $E_{\text{meas}}$
over all bias points. The bias with maximal achievable coupling rate is shown
as dotted dashed orange. The shading indicates the region of larger bias where
energy appears to be lost to other modes.

When the coupler is off, the high quality factor of the cavity should be
preserved by minimizing the loss introduced by the chip. In Fig. 2a, we
determine this loss by measuring the reflection coefficient $\Gamma$ from the
output port, using a signal that puts approximately 1000 photon into the
cavity. We sweep the bias close to critical coupling, where the reflection
amplitude is minimal and the corresponding linewidth is exactly twice the
internal loss rate. We observe that the total cavity linewidth is
$\kappa/2\pi=$ 3.46 $\pm$ 0.02 kHz, indicating the internal loss rate is
$\kappa_{\text{int}}/2\pi=$ 1.73 $\pm$ 0.01 kHz. Subtracting the 0.45 $\pm$
0.01 kHz loss rate of the cavity itself, measured separately without inserting
the chip, the introduced loss rate is about 1.28 $\pm$ 0.01 kHz.

When the coupler is on, the cavity field should couple strongly to the output
port, characterized by the maximal coupling rate $\kappa_{\text{max}}$. But in
order to detect if new loss channels are introduced by varying the bias, we
extract this maximal coupling rate from the measured decay rate $\kappa$ in
the time domain rather than the frequency domain. The protocol (as depicted in
the inset) is to drive the cavity at its resonance frequency with the coupler
on, so that the cavity field reaches a steady state, corresponding to
approximately $E_{\text{stored}}=8000$ photons (Fig. 2b). Then, both the drive
and the coupler are turned off for 1 $\mu$s to allow transients to settle.
Finally the coupler is turned back on at varying bias and the outgoing field
is measured. Because the energy in the cavity immediately before we turn on
the varying bias is kept constant, we can measure the portion of this energy
that reaches the output port and therefore identify additional loss channels.
Figure 2b shows three examples of time traces of the outgoing microwave field
amplitude at different biases $\Phi/\Phi_{0}$. Figure 2c shows the decay rate
as a function of bias extracted from this procedure and the measured energy of
the outgoing field. When the bias is tuned from under-coupled to over-coupled,
the external coupling becomes the dominant source of loss and the energy
plateaus. But at even higher bias, we observe a reduction in the measured
energy, so only the decay rate measured at the plateau can be interpreted as
the achievable external coupling rate. As a result, we identify
$\kappa_{\mathrm{max}}/2\pi=1.96\ \pm 0.06$ MHz. We define the on/off ratio as
$\kappa_{\mathrm{max}}/\kappa_{\mathrm{int}}=1130\pm 40$, because the residual
coupling when the coupler is off is much less than $\kappa_{\mathrm{int}}$
(see supplementary material).

Figure 3:  Measurements of coupler induced self-Kerr. We measure the cavity
frequency shift from its low power limit $\Delta$ at different signal power
(blue). A fit on the linear region is shown as the dashed orange line. At
higher power, the resonator becomes bistable, deviating from the linear
behavior.

To empty the cavity at the characteristic time of $1/\kappa_{\text{max}}$, it
must also be possible to turn the coupler on/off much faster than that.
Fitting the time domain traces in Fig. 2b, we find that a lower bound on the
switching speed is determined by the 50 MHz analog bandwidth of our analog-to-
digital converter. We model the time domain traces as decaying exponential
pulses that have been modified by a low pass filter with corner frequency
$\gamma_{c}/2\pi$, and fit to the function:

$V(t)\propto\Theta(t-t_{0})\frac{e^{-\gamma_{c}(t-t_{0})}-e^{-\kappa(t-t_{0})}}{\gamma_{c}-\kappa}$

where $t_{0}$ is the start time, $\kappa$ is the total decay rate, and
$\Theta(t-t_{0})$ is the unit step function. Although the rising edges in
figure 2b appear instantaneous, they contain two data points. Here we focus on
datasets at under-coupled biases, where the time traces are approximately step
functions, to reduce the fit sensitivity of $\kappa$ and improve that of
$\gamma_{c}$. We find $\gamma_{c}=2\pi\times(48\pm 5$ MHz)
$\gg\kappa_{\text{max}}$, indicating that the switching time is at most
$3.2\pm 0.4$ ns.

Finally, the coupler’s ability to isolate the cavity from other microwave
signals that might be present if this device were embedded in larger microwave
network is also characterized by the cavity frequency shift induced either by
an input signal from the measurement port or photons stored in the cavity. We
characterize this self-Kerr effect by measuring $\Gamma$ and extracting the
cavity frequency shift $\Delta$, caused by increasing the power in the probe
tone above its low power limit. We determine the resonance frequency in these
high-power reflection measurements as that frequency for which
$\partial\angle\Gamma(\omega)/\partial\omega$ is maximum. The result is shown
in figure 3. These data are measured with the coupler bias tuned to critical
coupling such that the number of photons $n$ in the cavity is
$n=P_{\text{in}}/(\kappa\hbar\omega)$, where $P_{\text{in}}$ is the incident
power at the output port, calibrated by dividing the attenuation of the lines
from the microwave power injected into the cryostat. Thus the Kerr shift can
also be expressed in terms of the number of cavity photons as $-0.04\pm 0.02$
Hz/photon. For operations using 1000 photons, the shift in the resonant
frequency is still small compared to a linewidth. This compares favorably to
other dynamic coupling implementations, which generally have $>1$ kHz/photon
self-Kerr Burkhart  _et al._ (2021); Pfaff  _et al._ (2017).

Table 1: Performance summary. loss and residual coupling | 1.28 $\pm$ 0.01 kHz
---|---
maximal coupling | 1.96 $\pm$ 0.06 MHz
on/off ratio | 1130 $\pm$ 40
switching time | < 3.2 $\pm$ 0.4 ns
self-Kerr | -0.04 $\pm$ 0.02 Hz/photon

In conclusion, we have demonstrated an interface integrating a loop antenna
and a coupling element that connects to a cavity, which has several desirable
properties as building blocks for modular quantum networks. To that end, we
envision extending this work by integrating multiple coupling elements onto
one chip and creating a small quantum network with a few cavities, made
monolithically, and connected by inserting such a chip into all of them
simultaneously. We anticipate that desirable features demonstrated in this
work can be preserved in such a scheme, indicating progress towards a larger,
modular quantum network.

See the supplementary material for the device photo and wiring diagram,
analysis on the coupling scheme, loop antenna design, uniform bias tuning
steps and transmission measurements.

This work was supported by Q-SEnSE: Quantum Systems through Entangled Science
and Engineering (NSF QLCI Award OMA-2016244) and the NSF Physics Frontier
Center at JILA (Grant No. PHY-1734006).

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# The quark orbital angular momentum of ground state octet baryons

Jing-Feng Li  School of Physical Science and Technology, Southwest University,
Chongqing 400715, China  Cheng Chen  Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 101408, China
Gang Li<EMAIL_ADDRESS>College of Physics and Engineering, Qufu Normal
University, Qufu 273165, China  Chun-Sheng An<EMAIL_ADDRESS>School of
Physical Science and Technology, Southwest University, Chongqing 400715, China
Cheng-Rong Deng<EMAIL_ADDRESS>School of Physical Science and Technology,
Southwest University, Chongqing 400715, China  Ju-Jun Xie
<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 101408, China  Lanzhou
Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu
Province, Lanzhou University, Lanzhou 730000, China

###### Abstract

Here we study the quark orbital angular momentum of the ground octet baryons
employing an extended chiral constituent quark model, within which the baryon
wave functions are taken to be superposition of the traditional $qqq$ and the
$qqqq\bar{q}$ higher Fock components. Coupling between the two configurations
is estimated using the ${}^{3}P_{0}$ quark-antiquark creation mechanism, and
the corresponding coupling strength is determined by fitting the sea flavor
asymmetry of the nucleon. The obtained numerical results show that the quark
angular momentum of the nucleon, $\Sigma$, $\Lambda$ and $\Xi$ hyperons are in
the range $0.10$-$0.30$. In addition, the quark angular momentum of all the
hyperons are a little bit smaller than that of the nucleon. And the octet
baryons spin fractions taken by the intrinsic quark orbital angular momentum
could be up to $60\%$ in present model.

##  I Introduction

In the traditional constituent quark models, the ground octet baryons are
composed of three quarks that are in their $S$-wave, thus the total spin of
the baryons are contributed from only the spin of their quark content, namely,
the $SU(2)$ mixed symmetric spin configuration of the three-quark system with
total spin $S=1/2$ should account for the total spin of corresponding baryons
completely Capstick:2000qj ; Glozman:1995fu . However, in the late 1980s, the
famous European muon collaboration (EMC) found that the quark spin could take
only a small fraction of the proton spin Ashman:1987hv ; Ashman:1989ig , which
raised up the proton spin crisis.

Since then intensively experimental measurements on the spin structure of
proton have been taken Anthony:2000fn ; COMPASS:2006mhr ; COMPASS:2010wkz ;
COMPASS:2015mhb ; COMPASS:2016jwv ; COMPASS:2017hef ; Deur:2021klh , and, on
the theoretical side, the spin decomposition of proton in gauge theory was
proposed in Refs. Ji:1996ek ; Balitsky:1997rs ; Ji:1997pf ; Chen:2008ag .
Accordingly, contributions of the quarks, the gluons and orbital angular
momentum (OAM) to the total proton spin have been investigated by lattice QCD
Aoki:1996pi ; Hagler:2007xi ; QCDSF:2011aa ; Yang:2016plb ; Alexandrou:2017oeh
; Yamanaka:2018uud ; Yang:2019dha ; RQCD:2019jai , chiral perturbation theory
Chen:2001pva ; Dorati:2007bk ; Lensky:2014dda ; Li:2015exr , as well as other
theoretical approaches Brodsky:1988ip ; Qing:1997th ; Qing:1998at ;
Brodsky:2000ii ; An:2005cj ; Brodsky:2006ha ; Barquilla-Cano:2006hku ;
Adamuscin:2007fk ; Myhrer:2007cf ; Thomas:2008ga ; Bijker:2009up ;
Lorce:2011kd , for recent reviews on status of the experimental and
theoretical investigations on nucleon spin structure, see Refs. Kuhn:2008sy ;
Burkardt:2008jw ; Leader:2013jra ; Wakamatsu:2014zza ; Liu:2015xha ;
Deur:2018roz ; Ji:2020ena .

Based on these previous theoretical studies, one may conclude that the
intrinsic sea content in baryons should play important roles in the static
properties of baryons. On the other hand, the experimentally observed sea
flavor asymmetry of proton NuSea:2001idv ; SeaQuest:2021zxb  also reveals the
non-perturbative effects of the intrinsic sea content in baryons.

Constructively, the proton can be depicted by a meson-cloud picture that the
proton is explained to be a neutron core surrounding by pion meson cloud.
Furthermore, one can also consider effects of the $K\Lambda$, $K\Sigma$, and
$\pi\Delta$ components in proton Ericson:1983um ; Thomas:1983fh ;
Henley:1990kw ; Brodsky:1996hc . Within the meson cloud model, spin of proton
could be directly decomposed into the spin and orbital angular momentum of
quarks, since the meson clouds should be in their $P$-wave states relative to
the baryon cores. And the sea flavor asymmetry may be described by the $\pi N$
and $\pi\Delta$ components with appropriate probabilities in proton. Within
this picture, one can also explain the intrinsic strange-antistrange quark
asymmetry in proton if the $K\Lambda$ and $K\Sigma$ components are taken into
account Brodsky:1996hc . Straightforwardly, the meson cloud picture for
nucleon can be extended to the other octet baryons Shao:2010wq .

Alternatively, the extended chiral constituent quark model (E$\chi$CQM), in
which the higher Fock components in the baryon’s wave function are assumed to
be compact pentaquark configurations, was proposed to study the strangeness
magnetic moment Zou:2005xy , strangeness form factor Riska:2005bh  and
strangeness spin of proton An:2005cj , and further developed to investigate
the intrinsic sea content and meson-baryon sigma terms of the octet baryons
An:2012kj ; An:2014aea ; Duan:2016rkr . In addition, the experimental data for
decays of baryon resonances such as $\Delta(1232)$, $P_{11}(1440)$,
$S_{11}(1535)$, $S_{11}(1650)$, $D_{13}(1520)$ and $D_{13}(1700)$ Li:2005jn ;
Li:2005jb ; Li:2006nm ; An:2008xk ; An:2009uv ; An:2011sb  can be also well
reproduced by E$\chi$CQM. Very recently, the E$\chi$CQM has been applied to
study the quark OAM of the proton and flavor-dependent axial charges of the
ground state octet baryons An:2019tld ; Wang:2021ild ; Qi:2022sus . It’s shown
that the singlet axial charge $g_{A}^{(0)}$, the isovector axial charge
$g_{A}^{(3)}$ and the $SU(3)$ octet axial charge $g_{A}^{(8)}$ of the octet
baryons obtained in the E$\chi$CQM, are consistent with predictions by lattice
QCD and chiral perturbation theory, if the model parameters are fixed by
fitting the data for $\bar{d}-\bar{u}$ asymmetry of proton SeaQuest:2021zxb .
As an direct extension of Ref. An:2019tld , here we investigate the quark OAM
of the ground octet baryons, and analyze the spin decomposition of the
corresponding baryons in the E$\chi$CQM.

The present manuscript is organized as follows. In Sec. II, we present the
framework which includes the E$\chi$CQM and the formalism for calculations on
the OAM in corresponding model, the explicit numerical results and discussions
are given in Sec. III. Finally, we give a brief summary of present work in
Sec. IV.

##  II Framework

To investigate the quark orbital angular momentum of the ground octet baryons,
here we employ the E$\chi$CQM, in which the wave functions of baryons can be
expressed in a general form as:

$|B\rangle=\frac{1}{\sqrt{\mathcal{N}}}\left(|qqq\rangle+\sum_{i}C_{i}^{q}|qqq(q\bar{q}),i\rangle\right),$
(1)

where the first term just represents the wave functions for the traditional
three-quark components of the octet baryons, while the second term denotes the
wave functions for the compact five-quark components, with the sum over $i$
runs over all the possible five-quark configurations with a $q\bar{q}$
($q=u,d,s$) pair that may form considerable higher Fock components in the
octet baryons. $C_{i}^{q}/\sqrt{\mathcal{N}}$ are the corresponding
probability amplitudes for the five-quark components with $\mathcal{N}$ being
a normalization constant. The seventeen possible $qqq(q\bar{q})$
configurations with $i=1\cdots 17$ are shown in Table 1, where the flavor,
spin, color and orbital wave functions for the four-quark subsystem are
denoted by the Young tableaux of the $S_{4}$ permutation group.

Table 1: The orbital-flavor-spin configurations for the four-quark subsystem of the five-quark configurations those may exist as higher Fock components in ground octet baryons. $i$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---
Config. | $[31]_{\chi}[4]_{FS}[22]_{F}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[211]_{F}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{1}}[22]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{2}}[22]_{S}$ | $[4]_{\chi}[31]_{FS}[211]_{F}[22]_{S}$
$i$ | $6$ | $7$ | $8$ | $9$ | $10$
Config. | $[4]_{\chi}[31]_{FS}[31]_{F_{1}}[22]_{S}$ | $[4]_{\chi}[31]_{FS}[31]_{F_{2}}[22]_{S}$ | $[31]_{\chi}[4]_{FS}[31]_{F_{1}}[31]_{S}$ | $[31]_{\chi}[4]_{FS}[31]_{F_{2}}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[211]_{F}[31]_{S}$
$i$ | $11$ | $12$ | $13$ | $14$ | $15$
Config. | $[31]_{\chi}[31]_{FS}[22]_{F}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{1}}[31]_{S}$ | $[31]_{\chi}[31]_{FS}[31]_{F_{2}}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[211]_{F}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[22]_{F}[31]_{S}$
$i$ | $16$ | $17$ |  |  |
Config. | $[4]_{\chi}[31]_{FS}[31]_{F_{1}}[31]_{S}$ | $[4]_{\chi}[31]_{FS}[31]_{F_{2}}[31]_{S}$ |  |  |

As we can see in Table 1, spin wave functions of the four-quark subsystem in
configurations with $i=1\cdots 7$ are $[22]_{S}$, which leads to the total
spin $S_{4}=0$ for the four-quark subsystem, thus one only need to consider
the combination of the quark (antiquark) OAM and spin of the antiquark to get
the total spin of a given baryon. And a general form for the wave functions of
these configurations can be expressed as

$\displaystyle|B,i=1\cdots 7\rangle_{5q}$ $\displaystyle=$
$\displaystyle\sum_{ijkln}\sum_{ab}\sum_{m\bar{s}_{z}}C^{\frac{1}{2},\uparrow}_{1,m;\frac{1}{2},\bar{s}_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}C^{[31]_{\chi
FS}^{k}}_{[O]_{\chi}^{i};[FS]_{FS}^{j}}C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[22]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle|I,I_{3}\rangle^{[F]_{F}^{l}}$
(2)
$\displaystyle|1,m\rangle^{[O]_{\chi}^{i}}|[22]_{S}^{n}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\}),$

with the coefficients $C^{[\cdots]}_{[\cdots][\cdots]}$ represent the CG
coefficients of the $S_{4}$ permutation group, and
$C^{\frac{1}{2},\uparrow}_{1,m;\frac{1}{2},\bar{s}_{z}}$ the CG coefficients
for the combination of the quark (antiquark) OAM and spin of the antiquark to
form a spin $|1/2,+1/2\rangle$ baryon state. And the explicit flavor, orbital,
spin and color wave functions for the presently considered five-quark
configurations have been given in Refs. Qi:2022sus .

While for the five-quark configurations with $i=8\cdots 17$ shown in Table 1,
the spin wave function of the four-quark subsystem is $[31]_{S}$ which results
in the total spin $S_{4}=1$ for the four-quark subsystem. Accordingly, we have
to take into account the combination of the spin of both the four-quark and
the antiquark and the quark (antiquark) OAM to get the total spin of the octet
baryon. One should note that combination of the spin for four-quark subsystem
$S_{4}=1$ and the quark OAM $L=1$ will lead to $J=L\oplus S_{4}=0$, $1$ or
$2$, and the former two $J$ those could form the total spin $S_{B}=1/2$ of the
presently studied octet baryons, when combine to the spin of the antiquark
$S_{\bar{q}}=1/2$ are applicable. Hereafter, we denote these two cases of wave
functions as Set I and Set II, respectively.

The general forms for wave functions of the five-quark configurations with
$i=8\cdots 17$ in these two cases are then given by

$\displaystyle|B,i=8\cdots 17\rangle_{5q}^{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\sum_{ijkln}\sum_{ab}\sum_{ms_{z}}C^{00}_{1,m;1,s_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}C^{[31]_{\chi
FS}^{k}}_{[O]^{\chi}_{i};[FS]_{FS}^{j}}C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[31]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle|I,I_{3}\rangle^{[F]_{F}^{l}}$
(3)
$\displaystyle|1,m\rangle^{[O]_{\chi}^{i}}|[31]_{S}^{n},s_{z}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\})\,,$

and

$\displaystyle|B,i=8\cdots 17\rangle_{5q}^{\mathrm{II}}$ $\displaystyle=$
$\displaystyle\sum_{ijkln}\sum_{ab}\sum_{J_{z}\bar{s}_{z}}\sum_{ms_{z}}C^{\frac{1}{2},\frac{1}{2}}_{1,J_{z};\frac{1}{2},\bar{s}_{z}}C^{1,J_{z}}_{1,m;1,s_{z}}C^{[1^{4}]}_{[31]_{\chi
FS}^{k};[211]_{C}^{\bar{k}}}C^{[31]_{\chi
FS}^{k}}_{[O]_{\chi}^{i};[FS]_{FS}^{j}}C^{[FS]_{FS}^{j}}_{[F]_{F}^{l};[31]_{S}^{n}}C_{a,b}^{[2^{3}]_{C}}|[211]_{C}^{\bar{k}}(a)\rangle|[11]_{C,\bar{q}}(b)\rangle$
(4)
$\displaystyle|I,I_{3}\rangle^{[F]_{F}^{l}}|1,m\rangle^{[O]^{\chi}_{i}}|[31]^{S}_{n},s_{z}\rangle|\bar{\chi},\bar{s}_{z}\rangle\phi(\\{\vec{r}_{q}\\})\,.$

Next we turn to the coefficients $C_{i}^{q}$ in Eq. (1). Generally, to get the
probability amplitude for a five-quark component in a given baryon, one has to
evaluate energy of the five-quark configuration $E_{i}^{q}$ and its coupling
to the three-quark component in the corresponding baryon.

In present work, the energy $E_{i}^{q}$ for a given five-quark configuration
is estimated using the chiral constituent quark model, within which the
hyperfine interaction between quarks is Glozman:1995fu

$\displaystyle H_{hyp}$ $\displaystyle=$
$\displaystyle-\sum_{i<j}\delta(r_{ij})\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\Bigg{[}\sum_{a=1}^{3}V_{\pi}(r_{ij})\lambda^{a}_{i}\lambda^{a}_{j}$
(5)
$\displaystyle+\sum_{a=4}^{7}V_{K}(r_{ij})\lambda^{a}_{i}\lambda^{a}_{j}+V_{\eta}(r_{ij})\lambda^{8}_{i}\lambda^{8}_{j}\Bigg{]}\,,$

where $\vec{\sigma}_{i(j)}$ and $\lambda^{a}_{i(j)}$ are the Pauli and flavor
$SU(3)$ Gell-Mann matrices acting on the $i(j)$-th quark, and $V_{M}(r_{ij})$
denotes the potential for exchanging a $M$-meson. Calculations on the matrix
elements $\langle qqq(q\bar{q}),i|H_{hyp}|qqq(q\bar{q}),i\rangle$ will lead to
the following common factors

$P^{M}_{l}=\langle lm|\delta(r_{ij})V_{M}(r_{ij})|lm\rangle\,,$ (6)

with $|lm\rangle$ the spatial wave function with OAM quantum number $l$. Here
we just take the empirical values for $P^{M}_{l}$ those could very well
reproduce the spectroscopy of light and strange baryons Glozman:1995fu :

$\displaystyle
P^{\pi}_{0}=29~{}\mathrm{MeV},~{}P^{K}_{0}=20~{}\mathrm{MeV},~{}P^{s\bar{s}}_{0}=14~{}\mathrm{MeV}\,,$
$\displaystyle
P^{\pi}_{1}=45~{}\mathrm{MeV},~{}P^{K}_{1}=30~{}\mathrm{MeV},~{}P^{s\bar{s}}_{1}=20~{}\mathrm{MeV}\,.$
(7)

Then the energy $E_{i}^{q}$ for the five-quark configurations listed in Table
1 can be calculated by

$E_{i}^{q}=E_{0}+\langle
qqq(q\bar{q}),i|H_{hyp}|qqq(q\bar{q}),i\rangle+n_{i}^{s}\delta m\,,$ (8)

where $E_{0}$ is a degenerated energy for all the studied configurations when
the hyperfine interaction between quarks and the flavor $SU(3)$ breaking
effects are not taken into account, and $\delta_{m}$ and $n_{i}^{s}$ denote
the mass difference of the light and strange quarks and number of strange
quarks in the corresponding five-quark system, respectively. Here both $E_{0}$
and $\delta_{m}$ are taken to be the empirical values An:2012kj :

$E_{0}=2127~{}\mathrm{MeV},~{}~{}~{}\delta_{m}=120~{}\mathrm{MeV}\,.$ (9)

Figure 1: Transition $qqq\rightarrow qqqq\bar{q}$ caused by a quark-antiquark
pair creation in a baryon via the ${}^{3}P_{0}$ mechanism.

To calculate the transition coupling matrix elements between the $qqq$ and
$qqqq\bar{q}$ components, here we adopt a widely accepted ${}^{3}P_{0}$ quark-
antiquark creation mechanism LeYaouanc:1972vsx ; LeYaouanc:1973ldf , which has
been used to study the intrinsic sea content of nucleon An:2012kj .
Explicitly, as depicted in Fig. 1, the three initial quarks go as spectators,
and a quark-antiquark pair with quantum number $J^{P}=0^{+}$ is created in the
vacuum, and then form the final five-quark system. The operator for this kind
of transition can be written as

$\displaystyle\hat{T}$ $\displaystyle=$
$\displaystyle-\gamma\sum_{j=1,4}\mathcal{F}_{j,5}^{00}\mathcal{C}_{j,5}^{00}\mathcal{C}_{OFSC}\sum_{m}\langle
1,m;1,-m|00\rangle\times$ (10)
$\displaystyle\chi_{j,5}^{1,m}\mathcal{Y}_{j,5}^{1,-m}(\vec{p}_{j}-\vec{p}_{5})b^{\dagger}(\vec{p}_{j})d^{\dagger}(\vec{p}_{5})\,,$

where $\gamma$ is an dimensionless transition coupling constant,
$\mathcal{F}_{j,5}^{00}\text{ and }\mathcal{C}_{j,5}^{00}$ are the flavor and
color singlet of the created quark-antiquark pair $q_{j}\bar{q}_{5}$,
$\chi_{j,5}^{1,m}\text{ and }\mathcal{Y}_{j,5}^{1,-m}$ are the total spin
$S_{q\bar{q}}=1$ and relative orbital $P-$ state of the created quark-
antiquark system, the operator $\mathcal{C}_{OFSC}$ is to calculate the
overlap factor between the residual three-quark configuration in the five-
quark component and the valence three-quark component, finally,
$b^{\dagger}(\vec{p}_{j}),d^{\dagger}(\vec{p}_{5})$ are the quark and
antiquark creation operators.

Then, one can calculate the probability amplitude for a five-quark
configuration $|qqq(q\bar{q}),i\rangle$ in a given baryon $B$ by the following
equation:

$C_{i}^{q}=\frac{\langle
qqq(q\bar{q}),i|\hat{T}|qqq\rangle}{M_{B}-E_{i}^{q}}\,,$ (11)

here $M_{B}$ denote the physical mass of the baryon $B$
ParticleDataGroup:2020ssz .

Explicit calculations on the transition matrix elements $\langle
qqq(q\bar{q}),i|\hat{T}|qqq\rangle$ between all the five-quark configurations
shown in Table 1 and the $qqq$ components in the presently studied octet
baryons will result in a common factor $\mathcal{V}$, namely,

$\langle qqq(q\bar{q}),i|\hat{T}|qqq\rangle=\mathcal{T}_{i}\mathcal{V}\,,$
(12)

the coefficient $\mathcal{T}_{i}$ can be obtained directly by the given wave
function of the $i$-th five-quark configuration, and $\mathcal{V}$ depends on
the ${}^{3}P_{0}$ transition coupling constant $\gamma$ and parameters of
explicit spatial wave functions determined by a given quark confinement
potential.

To reduce free model parameters, here we just fix $\mathcal{V}$ by fitting
experimental data for the intrinsic sea flavor asymmetry of the proton
SeaQuest:2021zxb

$I_{a}=\bar{d}-\bar{u}=\int_{0}^{1}[\bar{d}_{p}(x)-\bar{u}_{p}(x)]dx=0.118\pm
0.012\,.$ (13)

In present model, the $\bar{d}-\bar{u}$ asymmetry can be obtained by

$\displaystyle I_{a}$ $\displaystyle=$ $\displaystyle\bar{d}-\bar{u}$ (14)
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}}\Bigg{\\{}\Big{(}\frac{T_{1}^{l\bar{l}}}{M_{p}-E_{1}^{l}}\Big{)}^{2}+\Big{(}\frac{T_{11}^{l\bar{l}}}{M_{p}-E_{11}^{l}}\Big{)}^{2}+\Big{(}\frac{T_{15}^{l\bar{l}}}{M_{p}-E_{15}^{l}}\Big{)}^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{3}\bigg{[}\Big{(}\frac{T_{3}^{l\bar{l}}}{M_{p}-E_{3}^{l}}\Big{)}^{2}+\Big{(}\frac{T_{6}^{l\bar{l}}}{M_{p}-E_{6}^{l}})\Big{)}^{2}+\Big{(}\frac{T_{8}^{l\bar{l}}}{M_{p}-E_{8}^{l}}\Big{)}^{2}$
$\displaystyle+$
$\displaystyle\Big{(}\frac{T_{12}^{l\bar{l}}}{M_{p}-E_{12}^{l}}\Big{)}^{2}+\Big{(}\frac{T_{16}^{l\bar{l}}}{M_{p}-E_{16}^{l}}\Big{)}^{2}\bigg{]}\Bigg{\\}}\,,$

where $T_{i}^{l\bar{l}}$ denotes the transition coupling matrix element
between the $i$-th five-quark configuration in Table 1 with a light quark-
antiquark pair $l\bar{l}$ and the three-quark component in proton:

$T_{i}^{l\bar{l}}=\langle uud(l\bar{l}),i|\hat{T}|uud\rangle\,,$ (15)

and $E_{i}^{l}$ is energy of the corresponding five-quark configuration with a
light quark-antiquark pair.

Accordingly, the involved parameters in present model are the coupling
strengths $P_{l}^{M}$ (six involved ones in total), the degenerated energy
$E_{0}$ for all the five-quark configurations given in Table 1, mass
difference of the light and strange quark $\delta_{m}$, and the common factor
$\mathcal{V}$ for the transition matrix elements $\langle\hat{T}\rangle$. As
shown in Eqs. (7) and (9), the former eight parameters are taken to be the
empirical values used in the literature. And by fitting the sea flavor
asymmetry of proton in Eq. (13), one can get the following values of
$\mathcal{V}$:

$\displaystyle\mathcal{V}_{\rm I}=570\pm 46~{}~{}{\rm MeV}\,,$ (16)
$\displaystyle\mathcal{V}_{\rm II}=697\pm{80}~{}~{}{\rm MeV}\,,$ (17)

for the two different sets of wave functions using in present model,
respectively.

Note that one could also consider the higher Fock components
$qqq(q\bar{q})^{2}$ in the baryons. Energies of these kinds of components
those can couple to the ground state octet baryons may be about $600-900$ MeV
higher than the five-quark components, taking into account the two quark-
antiquark pairs creations, a rough estimation shows that the probability
amplitudes of the $qqq(q\bar{q})^{2}$ components are about $1/5$ of those of
the presently considered five-quark components.

Finally, within the E$\chi$CQM, the quark OAM of the octet baryons can be
calculated by

$L_{f}=\langle B|\hat{L}_{fz}|B\rangle=\frac{(C_{i}^{q})^{2}\langle
qqq(q\bar{q},i)|\hat{L}_{fz}|qqq(q\bar{q},i)\rangle}{\mathcal{N}}\,,$ (18)

with the operator $\hat{L}_{fz}$ defined by

$\hat{L}_{fz}=\sum_{f}(\hat{l}_{f}+\hat{l}_{\bar{f}})_{z}\,,$ (19)

where $\hat{l}_{f}$ and $\hat{l}_{\bar{f}}$ are the OAM operators for the
quark and antiquark with flavor $f$, respectively, and the sum runs over the
flavors $u$, $d$ and $s$.

##  III Numerical results and discussions

Figure 2: The probabilities of the five-quark components in the octet baryons, the boxes filled by white and grey colors are results obtained in Sets I and II, respectively, and the boxes with solid line border are results for the five-quark components with a light quark-antiquark pair, while those with dash line border are results for the five-quark components with a strange quark-antiquark pair. The rectangle height shows the uncertainty. Table 2: The quark OAM of the octet baryons. the upper and lower panels denoted by Set I and Set II are the numerical results obtained using the two sets of wave functions, respectively. |  | $p$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Lambda$ | $\Xi^{0}$
---|---|---|---|---|---|---
Set I | $L_{u}$ | $0.080(09)$ | $0.063(07)$ | $0.050(06)$ | $0.049(06)$ | $0.041(05)$
$L_{d}$ | $0.063(07)$ | $0.037(04)$ | $0.050(06)$ | $0.049(06)$ | $0.031(03)$
$L_{s}$ | $0.014(02)$ | $0.029(03)$ | $0.029(03)$ | $0.040(05)$ | $0.055(06)$
Set II | $L_{u}$ | $0.136(15)$ | $0.114(13)$ | $0.091(10)$ | $0.089(10)$ | $0.078(09)$
$L_{d}$ | $0.105(11)$ | $0.068(09)$ | $0.091(10)$ | $0.089(10)$ | $0.065(08)$
$L_{s}$ | $0.026(04)$ | $0.060(08)$ | $0.060(08)$ | $0.067(08)$ | $0.085(11)$

###  III.1 The quark OAM of the octet baryons

Within the framework of the E$\chi$CQM as shown in Sec. II, one can calculate
the probabilities of all the possible five-quark components in the octet
baryons, here we show the probabilities of the five-quark components with
light and strange quark-antiquark pairs in each octet baryon in Fig. 2. As
discussed explicitly in Qi:2022sus , the obtained baryon wave functions in the
both the Set I and Set II could result in the meson-baryon sigma terms
consistent with the predictions by other theoretical approaches, although the
probabilities for the five-quark components in the two Sets are very
different. And one may note that the upper limit of probabilities for the
intrinsic five-quark components obtained in Set II of present model are larger
than $50\%$, while these values are very close to those predicted in Refs.
Chang:2011vx ; Chang:2011du ; Chang:2014jba , where the model proposed by
Brodsky, Hoyer, Peterson and Sakai Brodsky:1980pb  was employed.

Using the wave functions shown in Eqs. (2), (3) and (4) with explicit
$C_{i}^{q}$ calculated by Eq. (11), one can directly calculate the quark OAM
of the octet baryons by Eq. (18). The corresponding numerical results for
$L_{u}$, $L_{d}$ and $L_{s}$ of the proton, $\Sigma^{+}$, $\Sigma^{0}$,
$\Lambda$ and $\Xi^{0}$ baryons are presented in Table 2. And the flavor-
dependent quark OAM of neutron, $\Sigma^{-}$ and $\Xi^{-}$ can be obtained
directly using the $SU(2)$ isospin symmetry, which yields

$\displaystyle L_{u(d)}(n)=L_{d(u)}(p)\,,~{}~{}~{}L_{s}(n)=L_{s}(p)\,,$ (20)
$\displaystyle
L_{u(d)}(\Sigma^{-})=L_{d(u)}(\Sigma^{+})\,,~{}~{}~{}L_{s}(\Sigma^{-})=L_{s}(\Sigma^{+})\,,$
(21) $\displaystyle
L_{u(d)}(\Xi^{-})=L_{d(u)}(\Xi^{0})\,,~{}~{}~{}L_{s}(\Xi^{-})=L_{s}(\Xi^{0})\,.$
(22)

And in Fig. 3, we depict the presently obtained total quark OAM $L_{q}$ of all
flavors compared to the results predicted by lattice QCD Alexandrou:2017oeh ,
unquenched quark model Bijker:2009up  and light cone constituent quark model
Lorce:2011kd .

Since all the valence quarks in the octet baryons are in their $S$-wave, the
quark OAM should be only contributed from the five-quark higher Fock
components of baryons. Therefore, the quark OAM should be sensitive to the
probabilities of the five-quark components. Therefore, as we can see in Table
2, the results for the quark OAM obtained in Set II model are about $1.5-2$
times of those in Set I model. On the other hand, in Set I, the five-quark
configurations with $i=8\cdots 17$ don’t contribute to the quark OAM of
proton, as discussed in Ref. An:2019tld , but it’s not true for Set II.
Consequently, the relationship between the flavor sea asymmetry and total
quark OAM of proton in Set II is

$L_{q}\approx 9/4I_{a}\,,$ (23)

instead of the $\sim 4/3I_{a}$ in Set I obtained in Ref. An:2019tld .

Figure 3: The total quark OAM of all flavors for the octet baryons. The
present results in Sets I and II are shown by (red) hollow and solid circles,
respectively. And the (blue) triangle, (purple) solid square and (green)
hollow square are results predicted by lattice QCD Alexandrou:2017oeh ,
unquenched quark model Bijker:2009up  and light cone constituent quark model
Lorce:2011kd , respectively.

For the nucleon, in both Sets I and II, the up and down quarks OAM are
comparable to each other, this is very different from the lattice QCD
predictions in Ref. Alexandrou:2017oeh , where the obtained light quark OAM
are $L_{u}=-0.107(40)$ and $L_{d}=0.247(38)$, while the total quark OAM of all
flavors in Ref. Alexandrou:2017oeh  is in the same range of the presently
obtained results, as shown in Fig. 3. In addition, the presently obtained
total quark OAM is in consistent with the predictions by unquenched quark
model Bijker:2009up  and light cone constituent quark model Lorce:2011kd .

For the other octet baryons, the light quarks OAM of the hyperons are smaller
than that of nucleon, while the strange quark OAM of hyperons are $2-4$ times
of that of nucleon. Obviously, this is because of the more strange quark
contents in hyperons than in nucleon. In Ref. Bijker:2009up , the total quark
OAM of $\Lambda$ was studied employing the unquenched quark model by
considering the effects of the quark-antiquark pairs created via the
${}^{3}P_{0}$ mechanism, and the predicted $L_{q}$ of $\Lambda$ hyperon is
$0.075$, that is smaller than the presently obtained results in both Sets I
and II.

###  III.2 The spin decomposition of the octet baryons

It’s also very interesting to investigate the spin decomposition of the octet
baryons. In present model, the spin and OAM of gluons are not involved,
accordingly, one can decompose the octet baryons spin by

$S_{B}=1/2=\sum_{f}\Big{[}\Big{(}\Delta f^{val}+\Delta
f^{sea}\Big{)}/2+L_{f}^{sea}\Big{]}\,,$ (24)

where $\Delta f^{val}/2$ and $\Delta f^{sea}/2$ are the flavor-dependent
valence and intrinsic sea quarks spin, respectively, and $L_{f}^{sea}$ the
intrinsic sea quark OAM which should be the same as those numerical results
shown in Table 2, and the sum over $f$ runs over the three flavors $u$, $d$
and $s$. We present our numerical results of Sets I and II obtained by
limiting the sea flavor asymmetry in proton to be the experimentally measured
central value $I_{a}=\bar{d}-\bar{u}=0.118$ SeaQuest:2021zxb  in Table 3,
compared to the results in the traditional three-quark constituent quark
model.

Table 3: Spin decomposition of the octet baryons. The three panels from left to right denoted by Traditional three-quark CQM, E$\chi$CQM Set I and E$\chi$CQM Set II are results in the traditional three-quark constituent quark model and the presently obtained results in Sets I and II, respectively. The last three rows denoted by $\mathcal{C}^{val}_{\Delta}$, $\mathcal{C}^{sea}_{\Delta}$, $\mathcal{C}^{\Sigma}_{\Delta}$ and $\mathcal{C}^{sea}_{L}$ are baryon spin fractions carried by the valence quark spin, intrinsic sea quark spin, total (valence and sea) quark spin and intrinsic sea quark OAM, respectively, in unit of $\%$. | Traditional three-quark CQM | E$\chi$CQM Set I | E$\chi$CQM Set II
---|---|---|---
| $p$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Lambda$ | $\Xi^{0}$ | $p$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Lambda$ | $\Xi^{0}$ | $p$ | $\Sigma^{+}$ | $\Sigma^{0}$ | $\Lambda$ | $\Xi^{0}$
$\Delta u^{val}$ | $4/3$ | $4/3$ | $2/3$ | $0$ | $-1/3$ | $0.839$ | $0.883$ | $0.442$ | $0$ | $-0.233$ | $0.606$ | $0.638$ | $0.319$ | $0$ | $-0.177$
$\Delta d^{val}$ | $-1/3$ | $0$ | $2/3$ | $0$ | $0$ | $-0.210$ | $0$ | $0.442$ | $0$ | $0$ | $-0.152$ | $0$ | $0.319$ | $0$ | $0$
$\Delta s^{val}$ | $0$ | $-1/3$ | $-1/3$ | $-1/3$ | $4/3$ | $0$ | $-0.221$ | $-0.221$ | $0.671$ | $0.933$ | $0$ | $-0.160$ | $-0.160$ | $0.497$ | $0.708$
$\Delta u^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0.044$ | $0.039$ | $0.031$ | $0.026$ | $0.018$ | $0.104$ | $0.124$ | $0.029$ | $-0.020$ | $-0.017$
$\Delta d^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-0.003$ | $0.023$ | $0.031$ | $0.026$ | $0.030$ | $-0.073$ | $-0.066$ | $0.029$ | $-0.020$ | $-0.059$
$\Delta s^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0.015$ | $0.016$ | $0.016$ | $0.003$ | $-0.003$ | $-0.020$ | $-0.020$ | $-0.020$ | $0.054$ | $0.090$
$L_{u}^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0.080$ | $0.063$ | $0.050$ | $0.049$ | $0.041$ | $0.136$ | $0.114$ | $0.091$ | $0.089$ | $0.078$
$L_{d}^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0.063$ | $0.037$ | $0.050$ | $0.049$ | $0.031$ | $0.105$ | $0.068$ | $0.091$ | $0.089$ | $0.065$
$L_{s}^{sea}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0.014$ | $0.029$ | $0.029$ | $0.040$ | $0.055$ | $0.026$ | $0.060$ | $0.060$ | $0.067$ | $0.085$
$\mathcal{C}^{val}_{\Delta}$ ($\%$) | $100$ | $100$ | $100$ | $100$ | $100$ | $63$ | $66$ | $66$ | $67$ | $70$ | $46$ | $48$ | $48$ | $50$ | $53$
$\mathcal{C}^{sea}_{\Delta}$ ($\%$) | $0$ | $0$ | $0$ | $0$ | $0$ | $6$ | $8$ | $8$ | $5$ | $5$ | $1$ | $4$ | $4$ | $1$ | $1$
$\mathcal{C}^{\Sigma}_{\Delta}$ ($\%$) | $100$ | $100$ | $100$ | $100$ | $100$ | $69$ | $74$ | $74$ | $72$ | $75$ | $47$ | $52$ | $52$ | $51$ | $54$
$\mathcal{C}^{sea}_{L}$ ($\%$) | $0$ | $0$ | $0$ | $0$ | $0$ | $31$ | $26$ | $26$ | $28$ | $25$ | $53$ | $48$ | $48$ | $49$ | $46$

As shown in Table 3, in the traditional three-quark constituent quark model,
only spin of the valence up and down quarks contribute to the nucleon spin.
While in the presently employed E$\chi$CQM, spin of the valence and intrinsic
sea quarks contribute $\sim 63\%$ and $6\%$ of the nucleon spin, respectively,
and the nucleon spin arisen from the intrinsic sea quark OAM is $31\%$, if the
wave functions of Set I are adopted. This is in good agreement with the
results in Bijker:2009up  obtained by the unquenched quark model. While if we
use the wave functions of Set II, the nucleon spin contributed from the
valence quark spin is $46\%$, and the intrinsic sea quark spin and OAM should
contribute $1\%$ and $53\%$ to the nucleon spin, respectively. In addition, if
the uncertainty in present model caused by fitting the sea flavor asymmetry
data are taken into account, contributions from the quark OAM to the nucleon
spin could be up to $60\%$.

Compared to the experimental data, i.e. the measured negative value for spin
of strange quark in proton $\Delta s$, and the proton spin fraction carried by
quark spin $\Sigma=0.33(10)$ Anthony:2000fn ; COMPASS:2006mhr ;
COMPASS:2010wkz ; COMPASS:2015mhb ; COMPASS:2016jwv ; COMPASS:2017hef ;
Deur:2021klh , one may conclude that the present Set II of wave functions
should be more favorable than Set I, although the value
$\mathcal{C}^{\Sigma}_{\Delta}=47\%$ for the quark spin fraction of proton
obtained in Set II model is still larger than experimental data.

The presently obtained quark spin fraction of the proton in the Set II model
is very close to the numerical result $0.46$ predicted in Barquilla-
Cano:2006hku  with contributions of only the one-body axial current. It’s
shown that the simple additive constituent quark model should violate the
partial conservation of the axial current (PCAC) condition Barquilla-
Cano:2002btm ; Barquilla-Cano:2003qmv , so one has to take into account
contributions of two-body axial exchange currents, which could deduce the one-
body axial current result by $\sim 40\%$ Barquilla-Cano:2006hku , and the
obtained quark spin fraction of proton including contributions from total
currents is consistent with the experimental data.

In Ref. Myhrer:2007cf , the authors employed a relativistic constituent quark
model, and took into account the one-gluon-exchange effects and the pion cloud
contributions, which resulted in that the quark OAM could contribute about
$62\%$ to the nucleon spin. This value is close to the presently obtained
upper limit for the quark OAM in the Set II model.

For the other octet baryons, as shown in Table 3, the baryon spin arisen from
the intrinsic sea quark spin should be less than $10\%$, while contributions
of the intrinsic sea quark OAM to the baryon spin are about $25-28\%$ and
$46-49\%$ using the wave functions of Set I and Set II, respectively. And one
should note that the contributions of the valence quark to spin of $\Sigma$,
$\Lambda$ and $\Xi$ hyperons in the present Set I model are $66\%$, $67\%$ and
$70\%$, respectively, while in Set II model, the corresponding contributions
are $48\%$, $50\%$ and $53\%$, respectively.

In Buchmann:2010pd ; Buchmann:2014ima , the quark spin contribution to the
total angular momentum of flavor octet and decuplet ground state baryons are
studied using a spin-flavor symmetry based parametrization method of quantum
chromodynamics. The results for the quark spin fraction of the ground state
octet baryons in Ref. Buchmann:2010pd  is $0.35(12)$, and the value obtained
in Ref. Buchmann:2014ima  is $0.41(12)$, which values are in general
consistent with the presently obtained results.

In Ref. Shanahan:2013apa , the MIT bag model with corrections from the one-
gluon-exchange and meson-cloud effects were employed to evaluate the octet
baryons spin fractions carried by the valence quarks, and the obtained results
were $42.6\%$, $58.9\%$ and $65.2\%$ for the $\Sigma$, $\Lambda$ and $\Xi$
hyperons, respectively. And in a very recent paper Suh:2022atr , the quark
spin content of $SU(3)$ light and singly heavy baryons were investigated using
a pion mean-field approach or the chiral quark-soliton model, flavor
decomposition of the axial charges of the baryons were studied explicitly, the
numerical results of the singlet axial charge for nucleon are comparable with
the present results in Set II model, within $1\sigma$, while those for the
other octet baryons are smaller than the present results. And it’s predicted
that the composite quark spin should contribute about $44\%$, $46\%$, $42\%$
and $41\%$ to spin of nucleon, $\Sigma$, $\Lambda$ and $\Xi$, respectively,
these values are also close to the present results in Set II.

##  IV Summary

To summarize, in present work, we investigate the quark orbital angular
momentum of the ground octet baryons employing the extended chiral constituent
quark model, in which the compact pentaquark higher Fock components in baryons
are taken into account. The probabilities of the higher Fock components are
determined by fitting the data for the sea flavor asymmetry of the proton.

In present model, there are two sets of wave functions for the pentaquark
higher Fock components in the ground octet baryons. These two sets of wave
functions could yield very different probability amplitudes of the pentaquark
components, as well different quark orbital angular momentum of the octet
baryons. Our numerical results show that the quark angular momentum of the
nucleon should be about $0.15-0.3$, namely, the quark angular momentum could
contribute $30-60\%$ to the nucleon spin. And the contributions of the
intrinsic sea quark spin to the nucleon spin are not large.

For the $\Lambda$, $\Sigma$, and $\Xi$ hyperons, it’s very similar to the
nucleon, the spin of the intrinsic sea quark contributes less than $10\%$ to
their total spin. And the obtained quark angular momentum are in the range
$0.10$-$0.28$, which are a little smaller than that of the nucleon. And the
corresponding total spin arisen from the quark orbital angular momentum is
about $25\sim 50\%$.

###### Acknowledgements.

This work is partly supported by the Chongqing Natural Science Foundation
under Project No. cstc2021jcyj-msxmX0078, and No. cstc2019jcyj-msxmX0409, and
the National Natural Science Foundation of China under Grant Nos. 12075288,
12075133, 11735003, 11961141012 and 11835015. It is also supported by the
Youth Innovation Promotion Association CAS, Taishan Scholar Project of
Shandong Province (Grant No.tsqn202103062), the Higher Educational Youth
Innovation Science and Technology Program Shandong Province (Grant No.
2020KJJ004).

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