# DME-Driver: Integrating Human Decision Logic and 3D Scene Perception in
Autonomous Driving

Wencheng Han  SKL-IOTSC, CIS, University of Macau  Dongqian Guo  SKL-IOTSC,
CIS, University of Macau  Cheng-Zhong Xu  SKL-IOTSC, CIS, University of Macau
Jianbing Shen Corresponding author SKL-IOTSC, CIS, University of Macau

###### Abstract

In the field of autonomous driving, two important features of autonomous
driving car systems are the explainability of decision logic and the accuracy
of environmental perception. This paper introduces DME-Driver, a new
autonomous driving system that enhances the performance and reliability of
autonomous driving system. DME-Driver utilizes a powerful vision language
model as the decision-maker and a planning-oriented perception model as the
control signal generator. To ensure explainable and reliable driving
decisions, the logical decision-maker is constructed based on a large vision
language model. This model follows the logic employed by experienced human
drivers and makes decisions in a similar manner. On the other hand, the
generation of accurate control signals relies on precise and detailed
environmental perception, which is where 3D scene perception models excel.
Therefore, a planning oriented perception model is employed as the signal
generator. It translates the logical decisions made by the decision-maker into
accurate control signals for the self-driving cars. To effectively train the
proposed model, a new dataset for autonomous driving was created. This dataset
encompasses a diverse range of human driver behaviors and their underlying
motivations. By leveraging this dataset, our model achieves high-precision
planning accuracy through a logical thinking process.

##  1 Introduction

Figure 1: Comparative Structures in Autonomous Driving Systems: (a) depicts an
planning-oriented autonomous driving system that optimizes for overall
performance using planning results, but lacks interpretability; (b) shows an
LLM-based autonomous driving system capable of producing reasonable control
signals, yet unable to fully leverage perception tasks; (c) illustrates our
DME-Driver Autonomous Driving System, which effectively strengths both
planning-oriented [25] and LVLM [35] models.

Autonomous driving systems represent a significant advancement in automotive
cars technology, combining computer vision and artificial intelligence. These
systems enable vehicles to perceive their surroundings [10, 41, 43, 51, 56],
make informed decisions [3, 18, 21, 29, 47], and navigate without human
intervention [27, 30, 37, 48, 54]. Key to these systems are sophisticated
perception mechanisms and decision-making algorithms. These components allow
vehicles to accurately understand their environment and make autonomous
decisions, ensuring safe and efficient navigation. However, the complexity of
these systems requires a focus on interpretability. Achieving interpretability
is not just a technical challenge, but also a crucial step towards wider
acceptance and integration of autonomous vehicles in society.

Recently, deep learning-based methods have achieved remarkable success in the
realm of autonomous driving [13, 17, 28, 34, 39]. Some works [11, 24, 20, 26]
proposed planning-oriented autonomous driving systems that can be trained end-
to-end. As illustrated in Fig. 1(a), this system [11] encompasses several
critical perception modules, including tracking, mapping, motion, and
occupancy detection. The outputs from these modules are fed into a planner,
which then generates the control signals for the vehicle. This approach
leverages the full potential of perception models in autonomous driving cars,
significantly enhancing the overall accuracy of the planning process. However,
the end-to-end nature of such methods leads to a lack of comprehensibility in
terms of human driving logic, resulting in a system that lacks
interpretability. When confronted with scenarios that the model fails to
resolve – what we might term ”bad cases” – it becomes challenging to
understand the flawed decision-making process of the system. This obscurity
poses a significant hurdle in troubleshooting and refining the autonomous
driving system, as there is little insight into why and how these errors are
made.

Benefiting from the advancements in large language models (LLMs), recent
approaches have attempted to enhance the interpretability of autonomous
driving systems by integrating LLMs into their frameworks [12, 33, 36, 42,
49], as shown in Fig. 1(b). Leveraging their formidable logic reasoning
capabilities and generalization, LLMs can effectively understand the
behavioral logic of human drivers in various driving scenarios. In the face of
corner cases, they can attempt to mimic human driver behavior, making reasoned
judgments and decisions. Moreover, even when decision-making errors occur, the
logical reasoning process of these models can be used as evidence to pinpoint
the causes of erroneous judgments and to seek solutions. However, these
methods place LLMs at the core of the autonomous driving system, treating the
outputs of perception modules as given conditions. This approach does not
fully utilize the diverse perception tasks essential for accurate
environmental perception. When perception models themselves produce erroneous
predictions, these mistakes can accumulate in the decision-making process. The
losses incurred from decision-making errors are then unable to retroactively
optimize the perception tasks, preventing the achievement of optimal overall
performance.

In this paper, we propose the DME (Decision-Maker Executor)-Driver autonomous
driving system, which synergizes the advantages of both LLMs and planning-
oriented perception models. Our system harnesses the LLMs’ ability to
understand human driver behavior logic while simultaneously leveraging the
precise environmental perception capabilities of planning-oriented autonomous
driving models. As depicted in Fig. 1(c), our system comprises two key roles:
the Decision-Maker and the Executor. The Decision-Maker is based on a Large
Vision Language Model (LVLM), trained through imitation learning on extensive
real-world driving data and human driver behavior logic. This enables the
Decision-Maker to thoroughly grasp the relationship between driving scenarios
and the underlying behavioral logic. When the vehicle encounters a new scene,
the Decision-Maker can simulate human-like logical assessments of key elements
in the scene, determining whether to accelerate, brake, or change lanes.
Additionally, the Decision-Maker can mimic a human driver’s ability to
describe key aspects of driving scenes and gaze information, providing the
perception model with a reliable set of prior information. This helps the
perception model to focus on elements of particular importance in the current
scenario.

The Executor, on the other hand, is responsible for converting the Decision-
Maker’s instructions into precise vehicle control signals. It acts as the
executor, ensuring that the high-level logic and reasoning from the Decision-
Maker are effectively translated into real-world driving actions. By doing so,
the Executor bridges the gap between decision-making and vehicle control,
enabling the system to navigate safely and efficiently in diverse driving
conditions. By fully considering these two aspects, DME-Driver aims to achieve
a more detailed and human-like understanding of driving scenarios, leading to
safer and more efficient autonomous driving decisions.

In summary, this paper presents four key contributions:

  * •

DME-Driver Autonomous Driving System: We present the DME-Driver system, which
combines the strengths of LLMs in logical reasoning and interpretability with
the precise environmental sensing of planning-oriented models, improving
decision-making robustness and interpretability in autonomous driving.

  * •

Human-Driver Behavior and Decision-Making (HBD) Dataset Leveraging both open-
source datasets and newly collected data, we developed a distinctive dataset
that integrates human driver behavior logic with detailed environmental
perception, specifically designed for training the DME-Driver system.

  * •

Decision-Maker Model Design: Our Decision-Maker model, based on LVLM, is
capable of imitating human driver instructions and focusing on important
elements in the environment, providing human-like insights for better
decision-making.

  * •

Executor Model Formulation: The Executor model accurately processes
environmental data and translates the Decision-Maker’s instructions into
accurate vehicle control signals, ensuring effective and context-aware
responses in various driving situations.

Empirical evaluation demonstrate that our method achieves state-of-the-art
accuracy in autonomous driving planning, significantly enhancing the system’s
interpretability. Every driving decision made by the system can be traced back
through logs to understand the underlying driving logic, providing a level of
transparency and explainability that is unprecedented in autonomous driving
systems.

##  2 Related Work

###  2.1 Autonomous Driving System

The development of autonomous driving technology has evolved from traditional
rule-based systems to sophisticated learning-based approaches. Initially,
autonomous vehicles were governed by algorithms that heavily relied on sensor-
based data and predefined rules [52, 40]. While these methods provided
reliable results in controlled environments, they struggled with the
unpredictability and complexity inherent in real-world driving conditions.

The advent of deep learning revolutionized this landscape, enabling systems to
learn from vast and varied datasets of real driving scenarios [1, 19]. This
shift to learning-based approaches has endowed autonomous systems with the
flexibility and adaptability needed to navigate complex and dynamic
environments more effectively, paving the way for more robust and versatile
driving systems.

A significant trend in recent years are the development of end-to-end
autonomous driving systems. These systems utilize deep neural networks to
process sensory inputs directly into driving actions, seeking to streamline
the autonomous driving process [5]. While this approach simplifies the system
architecture by eliminating modular decomposition, it raises challenges in
terms of interpretability and robustness. The black box nature of these
systems often hinders their ability to explain decisions and adapt to novel
situations, which is critical for ensuring safety and gaining user trust in
real-world applications.

###  2.2 Large Language Model for Autonomous Driving

The field of Large Language Models (LLMs) has seen remarkable growth,
significantly advancing capabilities in natural language understanding and
generation. Early models like BERT [14] and GPT [44] laid the groundwork for
more complex systems such as GPT-3 [7]. These models have not only excelled in
generating coherent text but have also shown proficiency in understanding
context and subtleties in language, making them invaluable for diverse
applications. A noteworthy advancement in LLMs is their integration with other
data modalities, particularly visual data. Models like CLIP [45] and DALL-E
[46] have demonstrated the effectiveness of combining textual and visual
information, enabling a holistic understanding of multimodal content. This
integration has been pivotal in broadening the applicability of LLMs to fields
where context and nuance across different data types are essential. However,
LLMs are not without their challenges. The computational requirements for
training and running these models are substantial, raising concerns about
environmental sustainability and the digital divide [50]. Furthermore, ethical
considerations, particularly regarding biases in model outputs and their
implications, are an ongoing area of concern and active research [4].

For a long time, autonomous driving systems have been treated as black boxes
with a lack of interpretability, making it difficult to understand how
decisions are made. The development of LLMs promises to solve this problem.
GPT-Driver [38] transforms the GPT-3.5 model into a motion planner for
autonomous driving, which demonstrates the motion planning abilities of the
LLMs. DriveGPT4 [55], an interpretable end-to-end autonomous driving system
based on LLMs, utilizes multimodal data such as videos, texts, and historical
control signals. It generates textual responses to questions and predicts
control signals for vehicle operation. Also, the reasoning abilities of LLMs
improve the performance in perception and understanding tasks. LLM-AD [16], a
semantic anomaly detection framework utilizing LLMs’ reasoning abilities,
demonstrates that the LLM-based monitor aligns with human intuition in both
fully end-to-end policies and classical autonomy stacks utilizing learned
perception.

##  3 Human Driver Behavior and Decision-Making Dataset

Figure 2: HBD Dataset Annotation Pipeline: There are four key steps in our
dataset creation - Data Collection, Prompt Design, Manual Corrections, and
Dialogue Generation, streamlining the process from raw data gathering to
structured dialogue formation.

###  3.1 Dataset Introduction

In order to thoroughly investigate the relationship between the behavioral
logic of human drivers and the resulting driving signals, our study focuses on
four key aspects. These aspects are essential for enhancing the robustness of
an autonomous driving system by understanding and mimicking human-like driving
behaviors.

Human Driver Gaze in Driving. Driving on roads requires real-time responses,
often based on subconscious reflexes rather than deliberate reasoning. To
comprehend these reflexive actions, we consider human gaze as an informative
behavioral signal. During driving, human drivers instinctively focus on the
most critical parts of the scene, which are typically directly influential in
the driving logic of the current scenario. These elements could include
traffic signals or other entities that might interact with the vehicle
shortly. Understanding this gaze behavior is crucial for our system to
recognize and prioritize important aspects of the driving environment.

Human Driver’s Understanding of Driving Scenes. The way human drivers
logically describe driving scenes provides a rich, purposeful understanding of
the scenario. Unlike standard image captions that offer a global view, human
drivers focus on elements and their interrelations that could impact driving
decisions. These logical descriptions enable even other human drivers to make
accurate judgments about appropriate driving actions. For instance, a
description might detail a scenario at an intersection where the vehicle is in
a left-turn lane with a red light for turning.

Decision-Making and Rationale of Human Drivers. The decision-making process of
human drivers is logical and information-rich. It encompasses the driver’s
synthesis of various factors to determine the appropriate action for a given
scenario, along with the underlying thought process. By comprehending and
emulating this aspect, an autonomous driving system can mimic human-like
decision-making logic. This capability is particularly valuable in novel or
challenging scenarios not encountered during training, enabling the system to
make safe and reliable judgments.

Precise Control Signals. The ultimate output of an autonomous driving system
should be structured control signals directly applicable to the vehicle.
Regardless of how correct and interpretable the natural language-based driving
logic is, it must be translated into concrete control commands. This
translation is vital to ensure that the detailed understanding and decisions
derived from human driving logic are effectively and safely executed by the
autonomous vehicle.

Given the requirement for four distinct types of raw data, we faced the
challenge that no existing open-source dataset comprehensively covers all
these aspects. To address this gap, we integrated three different open-source
datasets, each contributing one or more of the needed data types. For example,
the Look-Both-Ways [31] provides precise human driver gaze information
collected via eye-tracking devices. BDD-X [58] offers manually annotated
driving decisions and their underlying reasons. Nuscenes [8] includes detailed
control signals. Leveraging the powerful generalization capabilities of LVLM,
we can extrapolate the knowledge learned from each dataset to the others, even
in the absence of a single dataset containing all types of annotations.

Besides this, to further enhance the LVLM’s capabilities, especially in multi-
turn question-answering scenarios within the same scene, we collected a new,
comprehensive virtual sub-dataset named Virtual HBD. This dataset was gathered
using the Carla [15] simulation engine, where human drivers operated simulated
driving controls such as steering wheels and pedals. This process yielded
visual information and control signals for the vehicle. Additionally, drivers
manually annotated their specific decisions and the reasons behind them at
each moment. During the driving sessions, an eye tracker was employed to
capture the drivers’ gaze, focusing on the objects they observed in real-time.

Thanks to this novel Virtual HBD sub-dataset, we can link all related
information in a single dialogue, significantly bolstering the LVLM’s
generalization ability. This dataset uniquely enables the system to understand
and mimic the comprehensive decision-making process of human drivers, covering
everything from gaze patterns and scene interpretation to the rationale behind
specific driving decisions and their translation into control signals.

###  3.2 Data Collection and Labels Generation

To efficiently process and label this data, we employ a combination of
GPT-4V(ision) pre-annotation followed by manual corrections. Fig. 2 outlines
the data collection pipeline for our HBD Dataset. This pipeline comprises four
main steps, starting from data collection, involving Prompt Design and manual
correction, to the generation of dialogues.

Step 1: Data Collection As mentioned in the previous section, our data
collection integrates four sub-datasets, including three open-source datasets
- Look-Both-Ways [31], BDD-X [58], and Nuscenes [8] \- and one newly collected
sub-dataset, Virtual HBD. This comprehensive combination provides a rich base
of raw data encompassing various aspects of human driving behavior.

Step 2: Prompt Design To guide GPT-4V in analyzing driving behaviors from a
human driver’s perspective, we crafted unique prompts for each data type. For
example, with human gaze data, the aim is for LVLM to understand which areas a
human driver focuses on in a given scene and why. We first convert gaze points
into an axis-aligned bounding box by taking the minimum and maximum values of
x and y coordinates from the gaze points within 24 frames. This bounding box
is then drawn on the image. GPT-4V processes this enhanced image to infer the
driver’s focus and intentions.

Step 3: Manual Corrections After GPT-4V generates outputs, these are manually
reviewed and corrected. Completely incorrect responses are deleted and
regenerated with altered prompts. For correct responses, any details that do
not align with typical human driver thought processes are manually adjusted.

Step 4: Dialogue Generation Once all data are transformed into detailed
textual information, the next step is organizing this information into
dialogues, which involves three sub-steps:

First-person Conversion: All pronouns in the dialogues are converted to the
first person. This is to ensure that the subsequent LVLM Decision-Maker model
can process the dialogues from the perspective of the human driver.

Combining Multi-turn Dialogues: The labeling process typically involves
single-question prompts. For practical use, multi-turn dialogues offer more
contextual clues for the LVLM. We thus concatenate different types of Q&A
information into multi-turn dialogues, as shown in Fig. 2. The number of
dialogue turns varies – one to three turns for open-source data, depending on
the information available, and up to five turns for the Virtual HBD data.

Data Augmentation: To diversify the question-answer labels and enhance the
model’s generalization capabilities, we utilize GPT-3.5 to rewrite the
generated dialogues. This process involves changing the form of the dialogues
while keeping their content consistent.

##  4 The proposed DME-Driver Autonomous Driving System

In our DME-Driver Autonomous Driving System, as illustrated in Fig. 3, the
system is divided into two main components: the Decision-Maker and the
Executor. The Decision-Maker acts as the central decision-maker, synthesizing
vehicle status and current visual inputs to emulate a human driver’s logical
judgments. Its output is expressed in natural language, providing a logical
and interpretable narrative of driving decisions. This feature is particularly
valuable for diagnosing and understanding ’bad cases’ in driving scenarios, as
these natural language logs offer insights into the causes of incorrect
decisions. However, as natural language cannot directly control a vehicle, the
system incorporates the Executor network, functioning as a translator. This
network converts the Decision-Maker’s linguistic outputs into precise vehicle
control commands. The detailed architecture and functions of both the
Decision-Maker and the Executor networks are crucial to the system’s
effectiveness and will be elaborated in Sections 4.1 and 4.2 respectively.

Figure 3: The Detailed DME-Driver Autonomous Driving System: There are two
main components in our system. The Decision-Maker serves as the decision-
maker, emulating human drivers’ logic in various driving scenarios. The
Executor then translates the Decision-Maker’s instructions into precise
vehicle control signals, ensuring effective execution of driving decisions.

###  4.1 Decision-Maker: Human Driver Logic Understanding

The Decision-Maker in our DME-Driver system is a sophisticated Large Vision
Language Model (LVLM) designed to simulate the decision-making process of
human drivers. In our experiments, we utilize LLaVA [35] as the baseline
network for the Decision-Maker. This component is engineered to process inputs
from three different modalities: visual inputs from the current and previous
scenes, textual inputs in the form of prompts, and current status information
detailing the vehicle’s operating state.

Visual Input: To process the visual input, we utilize a pretrained CLIP [45]
visual encoder $E_{CLIP}$. This encoder converts the visual information into
feature tokens. To better comprehend the context of driving scenes, we enhance
the input by concatenating the previous three key frames with the current
frames into an image array:

$\text{F}_{v}=E_{CLIP}(F_{t-3}\oplus F_{t-2}\oplus F_{t-1}\oplus F_{t})$ (1)

Textual and Status Input: The prompt inputs $T_{p}$ and current status $T_{s}$
information are handled using a methodology similar to RT-2 [6], where a text
tokenizer is employed to encode these inputs uniformly. After encoding, the
tokens representing both visual and textual information are concatenated and
fed into the LLaMA 2 [53] model for processing:

$\displaystyle\text{F}_{t}$
$\displaystyle=\text{Tokenizer}(T_{p})\oplus\text{Tokenizer}(T_{s})$ (2) t
$\displaystyle=\text{LLaMA\\_2}(\text{F}_{t}+\text{F}_{v})$

This integrated approach allows the Decision-Maker to consider all aspects of
the driving scenario, ensuring a comprehensive understanding and simulation of
human-like decision-making processes. The final step involves a de-tokenizer,
which maps the output tokens back into natural language.

###  4.2 Executor: The Control Signal Generator

As depicted in Fig. 3, our Executor network in the DME-Driver system is
designed based on the UniAD [25] planning-oriented autonomous driving
framework, featuring 4 distinct components:

Backbone Network: The initial layer of the Executor network is a backbone
network, which is responsible for extracting features from multi-view vision
inputs. This network forms the foundation for subsequent feature processing
and interpretation. Following the backbone network, the extracted image
features are transformed into Bird’s Eye View (BEV) features through a process
similar to BEVFormer.

Perception Modules: The next stage consists of four specialized perception
modules:

TrackFormer is designed for detecting and tracking various elements within the
driving scene.

MapFormer generates a segmented map in BEV, providing detailed spatial
information about the environment.

MotionFormer predicts the motion trajectories of each element within the
scene.

OccFormer is responsible for generating occupancy information, indicating the
areas within the scene that are occupied and those that are free.

Planning Module: Following the perception modules, the Planning Module takes
the output tokens from these modules as its input. This module’s primary
function is to generate the predicted control signals for the vehicle.

Driver Logic Encoder: Distinct from the UniAD system, our Executor network
incorporates additional enhancements for the OccFormer and the Planning
module. The OccFormer combines textual information from scene descriptions and
gaze data, while the Planning module integrates decision-related text.
Specifically, we’ve integrated a Bert-based text encoder $E_{bert}$ to process
corresponding textual inputs:

$\displaystyle T_{occ}=E_{bert}(t_{gaze})\oplus E_{bert}(t_{description})$ (3)
$\displaystyle T_{planner}=E_{bert}(t_{decision}),$

where $T_{occ}$ represents the text encoding for the OccFormer, while
$T_{planner}$ represents the text encoding for the Planning module.
$t_{gaze}$, $t_{description}$, and $t_{decision}$ represent the answers
generated by the Decision-Maker for the Gaze, Scene Description, and Decision
Making questions, respectively. After encoding the text, we combine the BEV
feature $B$ generated by the backbone network with the corresponding text
encoding using a transformer fusion structure named LogicalFusioner. In this
structure, we consider the BEV feature as the query and the text encoding as
the key and value. After the aggregating of the multi-head attention, we add a
shortcut connection to the original $B$ and produce the enhanced BEV feature
$B^{\prime}$:

$\displaystyle B^{\prime}$ $\displaystyle=\text{LogicalFusioner}(B,T)$ (4)
$\displaystyle=\text{MHA}(Q=B,K=T,V=T)+B$

The inclusion of the TextEncoder in the Executor network enables it to go
beyond just processing visual information; it allows for the integration of
decision-making information and scene understanding provided by the Decision-
Maker, emulating human driver insights for more comprehensive and context-
aware driving decisions.

###  4.3 Training

The training of the DME-Driver system is streamlined into two essential steps.
Firstly, training the Decision-Maker involves using multi-type human driver
decision data to understand the human driving logic. Secondly, the training
focuses on the Executor, which is trained using Decision-Maker instructions,
perception labels, and control signals. By utilizing this data, the Executor
can learn how to accurately transform instructions into control signals.

Decision-Maker Training: The training of the Decision-Maker network in our
DME-Driver system encompasses two critical stages: pretraining and fine-
tuning. Initially, the model undergoes pretraining on diverse datasets,
including 593K image-text pairs from CC3M [9] and 100K video-text pairs from
WebVid-10M [2], focusing on general video-text alignment. This phase involves
training the video tokenizer while keeping the CLIP encoder and LLM weights
fixed. The fine-tuning stage then tailors the model to the specific needs of
interpretable autonomous driving. Here, the LLM is trained alongside the
visual tokenizer using 30K video-text pairs, from the proposed HBD Dataset,
and supplemented with 80K instruction-following image-text pairs from LLaVA
[35].

Executor Training: The training of the Executor component in our DME-Driver
system primarily follows the setup utilized by UniAD [25]. However, we
introduce specific modifications to enhance the system’s consistency.
Initially, similar to UniAD, we start by jointly training the perception
parts, namely the tracking and mapping modules, for six epochs. We then
proceed to an end-to-end training phase, which lasts for 20 epochs and
encompasses all perception, prediction, and planning modules. To ensure
alignment between the output signals of the planning module and the decisions
made by the Decision-Maker, we introduce an additional reinforcement learning
component during the training of the planning module. This component applies a
penalty whenever the control signals deviate from the Decision-Maker’s
decisions. Specifically, we category the Decision-Maker’s decisions into eight
distinct types, such as moving forward, turning left, turning right, among
others. For each of these decision types, we’ve established specific rules to
determine whether a given control signal corresponds to one of these
categories.

##  5 Experiment

###  5.1 Human Driver Logic

To assess the accuracy of the Decision-Maker in mimicking human driver
decision-making and judgment in driving scenarios, we conducted an evaluation
using the test set of the HBD dataset. Our primary goal was to determine
whether the Decision-Maker could accurately replicate the focus areas, scene
descriptions, reasoning, and decisions characteristic of human drivers.

In our evaluation process, we adapted a method similar to that used in
DriveGPT4, utilizing an advanced version of ChatGPT to generate assessment
metrics. Leveraging ChatGPT’s advanced reasoning capabilities, we designed it
to provide a numerical score ranging from 0 to 1 for each prediction, where a
higher score indicates better accuracy in mirroring human-like decision-
making. This scoring method allows for a nuanced and comprehensive assessment
of the Decision-Maker’s performance.

The evaluation of the Decision-Maker’s accuracy was conducted across four key
dimensions:

Gaze: Assessing the accuracy of the Decision-Maker in identifying areas of
focus during the driving process.

Scene Understanding: Evaluating how precisely the Decision-Maker describes the
elements present in the current driving scene.

Reasoning: Analyzing the correctness of the logic employed by the Decision-
Maker in making driving decisions.

Decision: Determining the accuracy of the final driving decisions made by the
Decision-Maker.

To benchmark our system’s performance, we compared the Decision-Maker’s
accuracy with that of other general-purpose large models, including LLaVA and
GPT-4V, in similar scenarios. The outcomes of these comparisons are detailed
in Table 1.

Table 1: Comparative Analysis of Driving Logic Understanding: This table contrasts our DME-Driver system with other large language models, emphasizing the proficiency in comprehending and interpreting driving logic. Method | Gaze | Scene Understanding | Logic
---|---|---|---
ChatGPT4$\uparrow$ | ChatGPT4$\uparrow$ | ChatGPT4$\uparrow$
LLaVA-7B[35] | 45.3 | 60.1 | 45.7
LLaVA-13B[35] | 48.3 | 65.2 | 48.2
GPT-4V[57] | 75.3 | 79.2 | 65.4
DME-Driver | 85.2 | 86.5 | 80.3

###  5.2 Planning

The aim of this experiment is to validate the accuracy of the entire DME-
Driver system in making autonomous driving decisions. Following methodologies
established in previous works, we focus on evaluating the final decision
accuracy of the system.

As shown in Table 2, the results of our experiments demonstrate that the DME-
Driver system successfully harnesses the logic-driven prompts from the
Decision-Maker, enhancing the decision-making precision of the planning
module. This integration not only leads to higher decision accuracy in diverse
driving situations but also maintains a detailed log of the decision-making
process. The ability to trace back and understand the rationale behind each
decision is a critical aspect of our system, adding a layer of
interpretability and accountability that is often lacking in autonomous
driving systems.

Table 2: Comparison of Planning Accuracy: This table showcases a comparative analysis between our DME-Driver system and state-of-the-art methods, highlighting the advancements in planning accuracy achieved by our approach. Method | Input | L2(m)$\downarrow$ | Col. Rate(%)$\downarrow$
---|---|---|---
1s | 2s | 3s | Avg. | 1s | 2s | 3s | Avg.
NMP[59] | Lidar | - | - | 2.31 | - | - | - | 1.92 | -
SA-NMP[59] | Lidar | - | - | 2.05 | - | - | - | 1.59 | -
FF[22] | Lidar | 0.55 | 1.20 | 2.54 | 1.43 | 0.06 | 0.17 | 1.07 | 0.43
EO[32] | Lidar | 0.67 | 1.36 | 2.78 | 1.60 | 0.04 | 0.09 | 0.88 | 0.33
ST-P3[23] | Vision | 1.33 | 2.11 | 2.90 | 2.11 | 0.23 | 0.62 | 1.27 | 0.71
UniAD[25] | Vision | 0.48 | 0.96 | 1.65 | 1.03 | 0.05 | 0.17 | 0.71 | 0.31
DME-Driver | Vision | 0.45 | 0.91 | 1.58 | 0.98 | 0.05 | 0.15 | 0.68 | 0.29
Table 3: Ablation Study Results: This table presents the impact of various components within our DME-Driver system, illustrating how each part contributes to the overall effectiveness and decision-making accuracy. Module | L2(m)$\downarrow$ | Col.Rate(%)$\downarrow$
---|---|---
Executor | 1.03 | 0.31
GT+Executor | 0.94 | 0.28
Decision-Maker + Executor | 0.96 | 0.28
Decision-Maker + Executor + CL | 0.98 | 0.29

###  5.3 Ablation Study

In our ablation study of the DME-Driver system, we methodically dissected the
impact of each component on decision-making effectiveness, as shown in Table
3. We began by assessing the standalone performance of the Executor without
Decision-Maker guidance, establishing a baseline. Next, we evaluated the
impact of substituting the Decision-Maker’s guidance with ground truth
language cues, observing potential improvements. Following this, we examined
the combined performance of the Decision-Maker and Executor, gauging their
collaborative efficiency. A crucial addition was the implementation of a
consistency loss mechanism, slightly reducing performance metrics but
significantly enhancing decision alignment between Executor and Decision-
Maker.

##  6 Conclusion

In addressing the challenges of interpretability and insufficient use of human
driver behavior patterns in autonomous driving systems, this paper introduces
the DME-Driver Autonomous Driving System, a novel framework comprising two
integral components: the Decision-Maker and the Executor. The Decision-Maker
serves as the central decision-maker, adeptly understanding and emulating
human driver logic, thus ensuring each action taken by the system is both
logical and accountable. The Executor complements this by effectively
translating the nuanced decisions of the Decision-Maker into precise vehicle
control signals, harnessing the strengths of perception tasks and planning
algorithms. To facilitate comprehensive training and understanding of human
driver behavior, we developed the HBD dataset, rich in diverse and essential
driving information such as gaze, decision logic, and operational signals. Our
empirical tests showcase the system’s capability to accurately replicate human
driver reasoning and actions. Combined with the Decision-Maker’s guidance, the
Executor successfully converts these into operational commands, elevating the
overall decision-making efficacy to a state-of-the-art level. This achievement
not only demonstrates the effectiveness of the DME-Driver system but also
marks a significant leap forward in the field of autonomous driving
technology.

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2305.19388

# A closer look at the Yukawa’s interaction from a symmetry group perspective

Luiz L. Lopes<EMAIL_ADDRESS>Centro Federal de Educação Tecnológica de
Minas Gerais Campus VIII, Varginha/MG, CEP 37.022-560, Brazil

###### Abstract

I investigate the use of the SU(3) Clebsch-Gordan coefficients in light of the
relations of completeness and closure. I show that in the case of
$\alpha_{V}=F/(F+D)~{}\neq$ 1, there is an additional interaction: the
exchange of a $\rho$ meson between a $\Lambda$ and a $\Sigma^{0}$ hyperon that
only affects the symmetric coupling. I then calculate these additional
coupling constants and show that this recovers the completeness and closure of
the SU(3) Clebsch-Gordan coefficients for all values of $\alpha_{V}$. Besides,
it increases the symmetry of the theory, once now we can group the baryon
octet into four doublets. Finally, I add the new coupling constants to study
numerical results in the hyperon onset in dense nuclear matter assuming
$\alpha_{V}$ as a free parameter.

D01, D06, D41, E32

##  1 Introduction

The study of nuclear physics is almost a century old. And despite its
senility, some techniques developed in the early years are still helpful today
in describing strongly interacting matter. In 1935, H. Yukawa Yukawa  proposed
that the interaction between nucleons was mediated by an exchange of massive
particles. Nowadays, such interaction is called a one-boson exchange, or
Yukawa coupling Bethe1951 , and it is expressed as the so-called Yukawa
Lagrangian:

$\mathcal{L}_{YUK}=-g_{BBM}(\bar{\psi}_{B}\psi_{B})M.$ (1)

The theory of strong force and the use of the Yukawa couplings had a great
leap with the works of J. Schwinger SCHWINGER1957  and especially with the
elegant and imperative work of J. J. Sakurai SAKURAI1960 . Based on current
conservations and local gauge invariance, Sakurai proposed a model that deals
explicitly with baryon-baryon interaction via vector mesons exchange. In such
a model, the $\omega$ meson couples to the hypercharge while the $\rho^{0}$
meson couples to the isospin.

With the development of symmetry group theories, Sakurai’s theory was
relegated as just a particular case of the more powerful and well-accepted
flavor SU(3) symmetry group theory Gell1962 ; Beh1962 ; Swart1963 ; DOVER1984
. However, with the onset of the more restrictive flavor-spin hybrid SU(6)
group: SU(6) $\supset$ SU(3) $\otimes$ SU(2), Sakurai’s theory was restored in
its full glory; and again, the $\omega$ meson couples to the hypercharge and
the $\rho$ meson couples to the isospin DOVER1984 ; Pais1964 ; Pais1966 ;
NAGELS1979 .

Although the Yukawa coupling explicitly deals with baryon-baryon interaction
via one-boson exchange, such interaction has proven extremely useful also in
many-body theories. In 1974, J. D. Walecka applied the Yuakwa coupling to
describe dense nuclear matter in mean field approximation (MFA) WALECKA1974 .
In this approach, the mesonic fields are replaced by their expected values and
the nucleons do not interact with each other but instead, they behave like a
free Fermi gas with a classical background field. The Walecka model and its
extensions are today known as quantum hadrodynamics Serot_1992  and soon
become a standard effective field theory to describe dense nuclear matter.

From the early 1990s on, the interest in studying neutron stars with exotic
matter has increased significantly, and to reduce the huge uncertainties about
the hyperon-meson coupling constants, the use of the SU(6) symmetry group
became a standard approach and is widely used, even in nowadays ELLIS1991 ;
SCHAFFNER1994 ; Schaffner1996 ; Pal1999 ; Schaffner2000 ; WEISSENBORNNPA ;
Tolos2017 ; Lopes2020EPJA . However, the discovery and confirmation of
hypermassive neutron stars in the earlier 2010s have shaken our trust in SU(6)
coupling constants. For instance, the J0348+0432 with a mass range of $2.01\pm
0.04~{}M_{\odot}$ Antoniadis  and especially the PSR J0740+6620, whose
gravitational mass is 2.08 $\pm$ 0.07 $M_{\odot}$ Miller2021 ; Riley2021
bring great tension between the onset of energetically favorable hyperons and
its well-known softening of the equation of state (EoS). This phenom is called
the hyperon puzzle. Quickly, several authors realized that it was possible to
reconcile massive neutron stars with hyperons in their core by partially
breaking the SU(6) symmetry in favor of the less restrictive flavor SU(3)
symmetry WeissPRC2012 ; WeissPRC2014 ; Miyatsu2013 ; Gusakov2014 ; Micaela2016
; Li2018 ; lopesnpa ; Dex2021 ; Lopes_2022 ; Lopes2022CTP ; HongCPC ; mateus ;
FU2022 ; lopesPRD ; Sedrakian2023 .

Although in the SU(3) the $\rho^{0}$ meson does not necessarily couples direct
to the isospin, its sign depends on the isospin projection Swart1963 ;
DOVER1984 . This implies that the coupling of the $\rho$ between the neutrons
is the opposite of those between the protons. The same is true for the $\Xi$’s
and for the $\Sigma$’s. Such behavior is summarized in Eq. 2.

$\displaystyle g_{nn\rho}=-g_{pp\rho},\quad
g_{\Xi^{-}\Xi^{-}\rho}=-g_{\Xi^{0}\Xi^{0}\rho},$ $\displaystyle
g_{\Sigma^{-}\Sigma^{-}\rho}=-g_{\Sigma^{+}\Sigma^{+}\rho}.$ (2)

Moreover, as someone can correctly guess, the coupling constant between
$\Lambda$’s and between $\Sigma^{0}$’s are null:

$\displaystyle g_{\Lambda\Lambda\rho}=g_{\Sigma^{0}\Sigma^{0}\rho}=0,$ (3)

once their isospin projection is zero.

When we are dealing with the Yukawa coupling (Eq; 1), especially in quantum
hadrodynamics, we usually assume that Dirac field $\bar{\psi}_{B}$ is the
complex conjugate of the field $\psi_{B}$. From the SU(3) point of view, that
is almost always true. Most of the $g_{BBM}$ is zero for crossed terms -i.e.;
if $\bar{\psi}_{B}$ and $\psi_{B}$ are not complex conjugates to each other.

The KEY point of the present work is that if we assume that $\bar{\psi}_{B}$
and $\psi_{B}$ are always complex conjugates to each other, the relation of
completeness and closure of the SU(3) Clebsch-Gordan coefficients is violated
if $\alpha_{V}$ $\neq$ 1\. This implies that, in this case, the set of
coupling constants is incomplete. Indeed, there are crossed Yukawa couplings
(sometimes called coupled channels):

$-g_{\Sigma^{0}\Lambda\rho}(\bar{\psi}_{\Sigma^{0}}\psi_{\Lambda})\rho^{0},\quad\mbox{and}\quad-
g_{\Lambda\Sigma^{0}\rho}(\bar{\psi}_{\Lambda}\psi_{\Sigma^{0}})\rho^{0},$ (4)

that may in fact differ from zero. From the field theory point of view
SCHWINGER1957 , the Eq. 4 indicates that the $\Sigma^{0}$ and the $\Lambda$
interact with each other via $\rho$ meson exchange. However, in the mean-field
approximation, the $\Lambda$ and the $\Sigma^{0}$ now interact with the
background field of the meson $\rho$. The strength of this interaction depends
only on the coupling constant.

In this work, I calculate the crossed coupling constants from Eq. 4 by
imposing that the Yukawa Lagrangian (Eq. 1) is invariant under the SU(3)
flavor symmetry group and show that this crossed coupling contributes to the
symmetric coupling while having no effect in the antisymmetric one. Therefore,
it restores the relation of completeness and closure for the symmetric
coupling, and as a consequence, for all values of $\alpha_{V}$. Thereafter, I
explicitly add the crossed Yukawa terms to build a more complete QHD
Lagrangian. Then, I calculate the new energy eigenvalues for the $\Lambda$ and
$\Sigma^{0}$ hyperons. Finally, we see how the modified energy eigenvalues
affect some of the microscopic and macroscopic properties in neutron stars and
dense nuclear matter assuming $\alpha_{V}$ as a free parameter.

##  2  The SU(3) Group Formalism

In the SU(3) symmetry group formalism (see ref. Swart1963 ; DOVER1984 ;
Pais1964 ; Pais1966 ; lopesPRD  and the references therein to additional
discussion), each eigenstate can be labeled as $|N~{}Y~{}I~{}I_{3}\rangle$,
where $N$ is the dimension of the representation, $Y$ is the hypercharge, $I$
is the total isospin and $I_{3}$ is the isospin projection. Assuming that the
Yukawa coupling of the QHD (Eq. 1) is invariant under the SU(3) flavor
symmetry group, implies that its eigenstate is $|0~{}0~{}0~{}0\rangle$, or
simply a unitary singlet.

The eigenstate of the $\rho^{0}$ is $|8~{}0~{}1~{}0\rangle$. Therefore, in
order to produce a Yukawa Lagrangian that is a unitary singlet, the direct
product ($\bar{\psi}_{B}~{}\otimes~{}\psi_{B}$) also must have the same
eigenstate: $|8~{}0~{}1~{}0\rangle$. As the hypercharge and isospin projection
are additive numbers, the simplest way to couple
($\bar{\psi}_{B}~{}\otimes~{}\psi_{B}$) to result in $|8~{}0~{}1~{}0\rangle$
is to assume that $\bar{\psi}_{B}$ and $\psi_{B}$ are complex conjugates to
each other. After that, we must couple the resulting $|8~{}0~{}1~{}0\rangle$
state to the $\rho^{0}$ meson in order to obtain the unitary singlet:
($\bar{\psi}_{B}~{}\otimes$ $\psi_{B}$) $\otimes~{}\rho^{0}$ =
$|0~{}0~{}0~{}0\rangle$. From the use of the Speiser method Swart1963 , there
are two ways to couple ($\bar{\psi}_{B}~{}\otimes~{}\psi_{B}$) to result in
the $|8~{}0~{}1~{}0\rangle$ state, typically, the antisymmetric and the
symmetric coupling. Therefore, the Yukawa Lagrangian of Eq. 1 can be rewritten
as:

$\mathcal{L}_{\rm Yukawa}=-((gC_{8}+g^{\prime}C^{\prime}_{8})\times
C_{1})(\bar{\psi}_{B}\psi_{B})\rho^{0},$ (5)

The $g$ ($g^{\prime}$) is the constant associated with the symmetric
(antisymmetric) coupling, while the $C_{8}$ ($C^{\prime}_{8}$) is the SU(3)
Clebsch-Gordan (CG) coefficients of the symmetric (antisymmetric) coupling to
result in the $|8~{}0~{}1~{}0\rangle$ state. Furthermore, $C_{1}$ is the CG
coefficients to the product $(\bar{\psi}_{B}\psi_{B})\times\rho^{0}$ to result
in the unitary singlet. The SU(3) CG coefficients can be calculated from the
isoscalar factors, as discussed in Ref. Swart1963 . Once its values are well
known, we use the tables presented in Ref. CG . Explicitly, we have:

$\displaystyle g_{pp\rho}={}$
$\displaystyle-\left(-\sqrt{\frac{3}{20}}g-\sqrt{\frac{1}{12}}g^{\prime}\right)\times\sqrt{\frac{1}{8}},$
$\displaystyle g_{nn\rho}={}$
$\displaystyle-\left(-\sqrt{\frac{3}{20}}g-\sqrt{\frac{1}{12}}g^{\prime}\right)\times-\sqrt{\frac{1}{8}},$
$\displaystyle g_{\Lambda\Lambda\rho}={}$
$\displaystyle-\left(0g+0g^{\prime}\right)\times 0,$ $\displaystyle
g_{\Sigma^{0}\Sigma^{0}\rho}={}$
$\displaystyle-\left(0g+0g^{\prime}\right)\times 0,$ $\displaystyle
g_{\Sigma^{+}\Sigma^{+}\rho}={}$
$\displaystyle-\left(0g-\sqrt{\frac{1}{3}}g^{\prime}\right)\times\sqrt{\frac{1}{8}},$
$\displaystyle g_{\Sigma^{-}\Sigma^{-}\rho}={}$
$\displaystyle-\left(0g+\sqrt{\frac{1}{3}}g^{\prime}\right)\times\sqrt{\frac{1}{8}},$
$\displaystyle g_{\Xi^{0}\Xi^{0}\rho}={}$
$\displaystyle-\left(-\sqrt{\frac{3}{20}}g+\sqrt{\frac{1}{12}}g^{\prime}\right)\times-\sqrt{\frac{1}{8}},$
$\displaystyle g_{\Xi^{-}\Xi^{-}\rho}={}$
$\displaystyle-\left(-\sqrt{\frac{3}{20}}g+\sqrt{\frac{1}{12}}g^{\prime}\right)\times\sqrt{\frac{1}{8}},$
(6)

Nevertheless, the SU(3) CG coefficients, as their SU(2) counterparts (see for
instance chapter 3 in Sakurai’s classical book Sakuraibook ), must satisfy the
relations of completeness and closure. In other words, we must have: $\sum
C^{2}_{8}=\sum C^{\prime 2}_{8}=\sum C^{2}_{8}=1$. However, one can easily
check that:

$\left\\{\begin{array}[]{cc}C_{8}^{2}&=0.6\\\ C^{\prime 2}_{8}&=1\\\
C_{1}^{2}&=0.75.\end{array}\right.$ (7)

The results in Eq. 7 show us that the set of coupling constants presented in
Eq. 6 are complete for the antisymmetric coupling ($g^{\prime}$), but not
complete for the symmetric one ($g$). There are some additional
($\bar{\psi}_{B}~{}\otimes~{}\psi_{B}$) product that still results in the
$|8~{}0~{}1~{}0\rangle$ state, but are not complex conjugate to each other.
Indeed, the direct product
$\bar{\psi}_{\Sigma^{0}}~{}\otimes~{}\psi_{\Lambda}$, as well the
$\bar{\psi}_{\Lambda}~{}\otimes~{}\psi_{\Sigma^{0}}$ produce an eigenstate
$|8~{}0~{}1~{}0\rangle$. The coupling constants $g_{\Sigma\Lambda\rho}$ and
$g_{\Lambda\Sigma\rho}$ can be calculated with the SU(3) Clebsch-Gordan (CG)
coefficients:

$\displaystyle
g_{\Sigma^{0}\Lambda\rho}=-\bigg{(}-\sqrt{\frac{1}{5}}g+0g^{\prime}\bigg{)}\times\sqrt{\frac{1}{8}},$
$\displaystyle
g_{\Lambda\Sigma^{0}\rho}=-\bigg{(}-\sqrt{\frac{1}{5}}g+0g^{\prime}\bigg{)}\times\sqrt{\frac{1}{8}}.$
(8)

As can be seen, these crossed couplings are only non-null in the symmetric
coupling ($g$), as in the antisymmetric one ($g^{\prime}$), the set was
already complete. When we add these two additional coupling constants, we
recover the relations of completeness and closure: $\sum C^{2}_{8}=\sum
C^{\prime 2}_{8}=\sum C^{2}_{8}=1$, implying that we now have a complete set
of coupling constants in agreement with the SU(3) group for both: the
antisymmetric and symmetric couplings. Moreover, as can be seen, unlike the
cases of isospin doublets (as protons and neutrons; $\Xi^{0}$ and $\Xi^{-}$,
etc) the $g_{\Sigma\Lambda\rho}$ and $g_{\Lambda\Sigma\rho}$ are both
positives and not opposite to each other as the ones in Eq. 2. Now, following
ref. Swart1963  we introduce the coupling constants:

$\displaystyle
g_{8}=\frac{\sqrt{30}}{40}g+\frac{\sqrt{6}}{24}g^{\prime},\quad\mbox{and}\quad\alpha_{V}=\frac{\sqrt{6}}{24}\frac{g^{\prime}}{g8},$
(9)

which results in:

$\displaystyle
g_{\Sigma^{0}\Lambda\rho}=g_{\Lambda\Sigma^{0}\rho}=\frac{2}{3}\sqrt{3}g_{8}(1-\alpha_{V}),\quad\mbox{implying}$
$\displaystyle\frac{g_{\Sigma^{0}\Lambda\rho}}{g_{NN\rho}}=\frac{2}{3}\sqrt{3}(1-\alpha_{V}).$
(10)

Within the flavor SU(3) symmetry, we have in principle three free parameters:
$\alpha_{V}$, the ratio $z=g_{8}/g_{1}$, and the mixing angle $\theta_{V}$. -
see ref. DOVER1984 ; Micaela2016 ; lopesPRD  to additional discussion) When we
assume the SU(6) symmetry we have:

$\alpha_{V}=1.00,\quad z=\frac{1}{\sqrt{6}},\quad\theta_{V}=35.264,$ (11)

and the Sakrurai proposals SAKURAI1960  are restored: the $\rho$ meson couples
to the isospin, therefore $g_{\Sigma\Lambda\rho}$ = 0.

As $\alpha_{V}$ = $F/(F+D)$ is a weight factor for the contributions of the
antisymmetric $F$ (corresponding to $\\{8^{\prime}\\}$) and the symmetric $D$
(corresponding to $\\{8\\}$) couplings relative to each other, when we assume
$\alpha_{V}$ = 1, the symmetric couplings is neglect (g =0), therefore SU(3)
CG coefficients already form a complete set without the need of the
$g_{\Sigma^{0}\Lambda\rho}$ coupling.

However, if $\alpha_{V}~{}\neq~{}1$, the symmetric coupling is also taken into
account. Consequently the $g_{\Sigma\Lambda\rho}~{}\neq~{}0$ and these
interactions must be considered to account for the completeness of the theory.
The now complete set of coupling constants in agreement with the SU(3) theory
is presented in Tab. 1. These results are fully model-independent and can be
applied to a diversity of calculations in future works. It is worth to point
that $\alpha_{V}$ = 1 is still a legitimate choice and was used to reproduce
hyperon-nucleon scattering data Rijken2010 . The phenomenological necessity of
the $g_{\Lambda^{0}\Sigma\rho}$ coupling in the context of the hyperon-nucleon
scatterings remains unknown.

| $\alpha_{v}$
---|---
| 1.00 | 0.75 | 0.50 | 0.25
$g_{\Lambda\Lambda\omega}/g_{NN\omega}$ | 0.667 | 0.687 | 0.714 | 0.75
$g_{\Sigma\Sigma\omega}/g_{NN\omega}$ | 0.667 | 0.812 | 1.0 | 1.25
$g_{\Xi\Xi\omega}/g_{NN\omega}$ | 0.333 | 0.437 | 0.571 | 0.75
$g_{\Lambda\Lambda\phi}/g_{NN\omega}$ | -0.471 | -0.619 | -0.808 | -1.06
$g_{\Sigma\Sigma\phi}/g_{NN\omega}$ | -0.471 | -0.441 | -0.404 | -0.354
$g_{\Xi\Xi\phi}/g_{NN\omega}$ | -0.943 | -0.972 | -1.01 | -1.06
$g_{\Lambda\Lambda\rho}/g_{NN\rho}$ | 0.0 | 0.0 | 0.0 | 0.0
$g_{\Sigma\Sigma\rho}/g_{NN\rho}$ | 2.0 | 1.5 | 1.0 | 0.5
$g_{\Xi\Xi\rho}/g_{NN\rho}$ | 1.0 | 0.5 | 0.0 | -0.5
$g_{\Sigma^{0}\Lambda\rho}/g_{NN\rho}$ | 0.0 | 0.288 | 0.577 | 0.866
Table 1: Complete set of baryon-vector mesons coupling constants for different
values of $\alpha_{v}$, within the SU(3) symmetry group. These results are
fully model-independent.

##  3 The QHD Formalism and numerical results

I now study the effects of $g_{\Sigma^{0}\Lambda\rho}$ on dense nuclear matter
for $\alpha_{V}~{}\neq$ 1 and compare the results with those without this
term.

I began by imposing chemical equilibrium and zero electric charge net, a
situation expected in neutron star interiors, to investigate the influence of
the crossed terms. Let us start with a classical QHD Lagrangian without
crossed couplings. Its Lagrangian reads Miyatsu2013 ; lopesPRD :

$\displaystyle\mathcal{L}=\sum_{B}\bar{\psi}_{B}[\gamma^{\mu}(\mbox{i}\partial_{\mu}-g_{B\omega}\omega_{\mu}-g_{B\phi}\phi_{\mu}-g_{B\rho}\frac{1}{2}\vec{\tau}\cdot\vec{\rho}_{\mu})$
$\displaystyle-(M_{B}-g_{B\sigma}\sigma)]\psi_{B}-U(\sigma)+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{s}^{2}\sigma^{2})$
$\displaystyle-\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\frac{1}{2}m_{v}^{2}\omega_{\mu}\omega^{\mu}+\Lambda_{\omega\rho}(g_{\rho}^{2}\vec{\rho^{\mu}}\cdot\vec{\rho_{\mu}})(g_{\omega}^{2}\omega^{\mu}\omega_{\mu})$
$\displaystyle-\frac{1}{4}\Phi^{\mu\nu}\Phi_{\mu\nu}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}+\frac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\;\mu}-\frac{1}{4}\bf{P}^{\mu\nu}\cdot\bf{P}_{\mu\nu},$
(12)

in natural units. Additional discussion about the parameters and the formalism
can be found in ref. Miyatsu2013 ; lopesnpa ; WALECKA1974 ; Serot_1992  and
the references therein. The g’s in Eq. 12 have only two instead three
subscripts to let clear that in this Lagrangian $\bar{\psi}_{B}$ is always the
complex conjugate of $\psi_{B}$. Applying Euler-Lagrange and the quantization
rules we obtain the energy eigenvalues (which at T = 0 K is also the chemical
potential). In MFA we have:

$E_{B}=\sqrt{M^{*2}_{B}+k^{2}}+g_{B\omega}\omega_{0}+g_{B\phi}\phi_{0}+\frac{\tau_{3}}{2}g_{B\rho}\rho_{0}$
(13)

Now I add the coupled channels in the Lagrangian of Eq. 12:

$\mathcal{L}_{\Lambda\Sigma^{0}\rho}=-\frac{1}{2}g_{\Sigma\Lambda\rho}(\bar{\psi}_{\Lambda}\psi_{\Sigma}+\bar{\psi}_{\Sigma}\psi_{\Lambda})\rho_{0},$
(14)

where the $1/2$ factor was added to keep the internal coherence with Eq. 12.
When we apply Euler-Lagrange to now complete SU(3) Lagrangian, we see that the
energy eigenvalue for all other six baryons is kept as in Eq. 13. For the
$\Lambda$ and the $\Sigma^{0}$ we have two coupled equations:

$\begin{cases}[\gamma^{\mu}(\mbox{i}\partial_{\mu}-g_{\Lambda\omega}\omega_{\mu})-M^{*}_{\Lambda}]\psi_{\Lambda}-\frac{1}{2}(g_{\Sigma^{0}\Lambda\rho})\rho_{0}\psi_{\Sigma}=0\\\
[\gamma^{\mu}(\mbox{i}\partial_{\mu}-g_{\Sigma\omega}\omega_{\mu})-M^{*}_{\Sigma}]\psi_{\Sigma}-\frac{1}{2}(g_{\Lambda\Sigma^{0}\rho})\rho_{0}\psi_{\Lambda}=0.\end{cases}$
(15)

However, as we already know the energy eigenvalue without the coupled channel,
their inclusion is much easier in Hamiltonian formalism. The diagonal terms
are the well-known unperturbed energy eigenvalues given by Eq. 13, while the
crossed terms are off-diagonal. We have:

$H=\left(\begin{array}[]{cc}E_{B}&\Delta\\\
\Delta&E_{B}\end{array}\right)~{}\mbox{and}~{}H|\psi_{B}\rangle=E|\psi_{B}\rangle,$
(16)

where $|\psi_{B}\rangle=(\psi_{\Lambda},\psi_{\Sigma})$ and
$\Delta=1/2(g_{\Sigma^{0}\Lambda\rho})\rho_{0}$. As we are dealing with a
beta-stable matter, $\mu_{\Lambda}=\mu_{\Sigma}$, the new energy eigenvalues
are (see for instance chapter 5 of Sakurai’s book Sakuraibook  for a complete
discussion):

$\begin{array}[]{c}E_{1}=\sqrt{M_{\Lambda}^{*2}+k^{2}}+g_{\Lambda\omega}\omega_{0}+g_{\Lambda\phi}\phi_{0}-\frac{g_{\Sigma\Lambda\rho}}{2}\rho_{0},\\\
E_{2}=\sqrt{M_{\Sigma}^{*2}+k^{2}}+g_{\Sigma\omega}\omega_{0}+g_{\Sigma\phi}\phi_{0}+\frac{g_{\Sigma\Lambda\rho}}{2}\rho_{0}.\end{array}$
(17)

Despite the energy eigenvalues from Eq. 17 being exact, the issue here is that
the coupled channels lead us to mixed states PROVIDENCIA1985 . In other words,
the $\psi_{\Lambda}$ and $\psi_{\Sigma}$ are not eigenstates of the
Hamiltonian of Eq. 16 anymore. Instead, we have a superposition Sakuraibook ;
PROVIDENCIA1985 . However, as we have $E_{B}>>\Delta$ in Eq. 16, and following
Sakurai’s nomenclature Sakuraibook , $\psi_{\Lambda}$ and $\psi_{\Sigma}$ are
”almost good” eigenstates of Eq. 16. Therefore we can recognize $E_{1}$ as the
eigenvalue of the $\Lambda$ and $E_{2}$ as the eigenvalue of the $\Sigma^{0}$.

Reducing the coupled channel to MFA is not new. It was successfully used to
account for the kaon interaction in nuclear medium in MFA (see, for instance,
section 10.1 of Glendenning’s book Glenbook  and the references therein.),
though such interaction is explicitly a coupled channel coming from the
$g_{N\Lambda K}$ and $g_{N\Sigma K}$ couplings Weise1988 ; KOCHPLB1994
(indeed, as the $g_{\Lambda\Lambda\rho}$, the $g_{NNK}$ is null Swart1963 ).
It is also worth to point that the $\Lambda$-$\Sigma$ interaction is supported
by experimental data, in the so-called coherent $\Lambda-\Sigma$ coupling
Aka2000 ; Aka2002  Finally, the eigenvalues of the other six baryons are given
by their usual expression, Eq. 13.

It is interesting to notice that when I calculated the
$g_{\Sigma^{0}\Lambda\rho}$ and the $g_{\Lambda\Sigma^{0}\rho}$ coupling
constants from the SU(3) Clebsch-Gordan coefficients, I showed that both have
positive signs. However, as they are off-diagonal contributions, they
ultimately contribute with opposite signs to the energy eigenvalues, as
displayed in Eq. 17. So, for practical purposes, the ($\Sigma^{0},~{}\Lambda$)
forms a new isospin doubled, exactly as the (p,n),
($\Sigma^{+},~{}\Sigma^{-}$), and ($\Xi^{0},~{}\Xi^{-}$), with the coupling
constants given by Tab. 1. The total EoS is given by Miyatsu2013 :

$\displaystyle\epsilon=\sum_{B}\frac{1}{\pi^{2}}\int_{0}^{k_{Bf}}dkk^{2}\sqrt{k^{2}+M_{B}^{*2}}+U(\sigma_{0})+\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}$
$\displaystyle+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}+\frac{1}{2}m_{\rho}^{2}\rho_{0}^{2}+3\Lambda_{v}\omega_{0}^{2}\rho_{0}^{2}+\sum_{l}\frac{1}{\pi^{2}}\int_{0}^{k_{lf}}dkk^{2}\sqrt{k^{2}+m_{l}^{2}},$
(18)

where $B$ indicats baryon and $l$ indicates leptons. The pressure is easily
obtained by thermodynamic relations: $p=\sum_{f}\mu_{f}n_{f}-\epsilon$, where
the sum runs over all the fermions and $\mu_{f}$ is the corresponding chemical
potential.

To obtain numerical results, I consider $\alpha_{V}$ a free paramenter but use
only $\alpha_{V}$ = 0.25, which has the strongest influence of the
$g_{\Sigma^{0}\Lambda\rho}$, in order to not saturate the figures. Also, I use
two different parameterizations, the eL3$\omega\rho$ lopesPRD , that virtually
fulfill every constraint of the symmetric nuclear matter, and the well-known
and the widely used GM1 paramertrization GM1 . All parameters and predictions
for the eL3$\omega\rho$ are presented in Tab. I of ref. lopesPRD , while the
GM1 can be found in Tab. I of ref. lopesnpa . The coupling constants of the
hyperons with the scalar meson are fixed to reproduce the hyperon potential
depth values: $U_{\Lambda}$ = - 28 MeV and $U_{\Sigma}$ = + 30 MeV. For the
and $U_{\Xi}$, I chose $U_{\Xi}$ = -18 MeV as suggested in ref. Potentials
when I use the GM1 parametrization (which allows a direct comparison with the
results presented in ref. lopesnpa ), and chose $U_{\Xi}$ = - 4 MeV as
suggested in ref. LQCD  for the eL3$\omega\rho$ parametrization (which allow
us a comparison with the results presented in ref. lopesPRD  ).

The reason I use two different parametrizations is that in the eL3$\omega\rho$
there is a non-linear coupling between the $\omega$ and $\rho$ mesons, as
introduced in the IUFSU model IUFSU , while for the GM1 there isn’t. Such
coupling influences the mass of the $\rho$ meson, which ultimately affects the
strength of the $\rho$ field at high densities.

|
---|---
|
Figure 1: (Color online) Particle population for the eL$\omega\rho$ and for
the GM1. Results with (without) * indicate the presence (absence) of the
$g_{\Sigma^{0}\Lambda\rho}$ coupling.

The particle population for the beta-stable matter at T = 0 K for $\alpha_{v}$
= 0.25 is displayed in Fig. 1. We can see that the main effect of the
$g_{\Sigma^{0}\Lambda\rho}$ coupling is to suppress the $\Lambda$ onset,
pushing it away to higher densities, whilst, at the same time, it favors the
$\Xi^{-}$. In the case of the eL3$\omega\rho$ parametrization, the prensece of
the $g_{\Sigma^{0}\Lambda\rho}$ coupling, pushes the $\Lambda$ threshold from
0.4114 fm-3 to 0.4416 fm-3, whilst the $\Xi^{-}$ is draw close-approach from
from 0.5821 fm-3 to 0.5168 fm-3. This indicates an increase of around $10\%$
in the density of the $\Lambda$ and a decrease of around $10\%$ in the density
of the $\Xi^{-}$. In the case of the GM1 parametrizations, the results are
more extreme. The $g_{\Sigma^{0}\Lambda\rho}$ coupling not only suppresses the
$\Lambda$ threshold whilst favoring the $\Xi^{-}$, but it exchanges their
roles. Within it, the $\Xi^{-}$ is now the first hyperon to appear and becomes
the most populous hyperon at higher densities. The $\Lambda$ threshold is
pushed away from 0.3264 fm-3 to 0.4405 fm-3; an increae of around $35\%$. On
the other hand, the $\Xi^{-}$ is drawn close-approach from from 0.4079 fm-3 to
0.3655 fm-3, a decrease of around $10\%$.

Now I use the EoS of the beta-stable electric neutral matter to solve the TOV
equations TOV  equations. For both parametrizations, I use the BPS EoS BPS
for the outer crust and the BBP EoS BBP  for the inner crust. I do not plot
the EoS itself because the effects of the $g_{\Sigma^{0}\Lambda\rho}$ coupling
are visually indistinguishable. The numerical results are presented in Fig. 2.

We can also discuss some constraints related to neutron stars. Today, maybe
the more important constraint is the undoubted existence of supermassive
neutron stars. Using the NICER x-ray telescope, ref. Riley2021  was able to
constraint the mass and the radius of the PSR J0740 + 6620 in the range of
$M=2.08\pm 0.07M_{\odot}$, and 11.41 km $<~{}R~{}<$ 13.70 km respectively. We
plot this constraint as a hatched area in Fig. 2. As can be seen, both the
eL3$\omega\rho$ and the GM1 fulfill this constraint.

Other constraints are related to the radius and tidal parameter of the
canonical 1.4 $M_{\odot}$ star, however, they are still the subject of high
debate about their true values. Recently, results obtained from Bayesian
analysis indicate that the radius of the canonical star lies between 10.8 km
and 13.2 km Yuxi ; and 11.3 km to 13.5 km Michael ; whilst results coming from
the NICER x-ray telescope points out that $R_{1.4}$ lies between 11.52 km and
13.85 km from ref. Riley:2019yda  and between 11.96 km and 14.26 km from ref.
Miller:2019cac . State-of-the-art theoretical results at low and high baryon
density point to an upper limit of $R_{1.4}$ $<$ 13.6 km (Annala2, ). Finally,
PREX2 results (PREX2, ) indicate that the radius of the canonical star lies
between 13.25 km $<~{}R_{1.4}~{}<$ 14.26 km. I use the intersection between
the two NICER results Miller:2019cac ; Riley:2019yda : 11.96 km
$<~{}R_{1.4}~{}<$ 13.85 km as a constraint for the canonical star.

---
Figure 2: (Color online) Left: Neutron stars mass-radius relation for the
eL$\omega\rho$ and the GM1 models. The solid (dotted) lines indicate the
presence (absence) of the $g_{\Sigma^{0}\Lambda\rho}$ coupling. The orangey
hatched area is the mass-radius uncertainty of the PSR J0740+6620 pulsar
Riley2021 , and the bluish hatched area is the intersection of two estimative
from NICER for the 1.4$M_{\odot}$ Riley:2019yda ; Miller:2019cac . Right: Zoom
in for $M~{}\geq~{}2.0M_{\odot}$.

In relation to the tidal parameter, an upper limit of 860 was found in ref.
Michael . A close limit, $\Lambda_{1.4}~{}<$ 800 was pointed out in ref.
tidal4 . In ref. Yuxi , an upper limit of 686 was deduced from Bayesian
analysis. On the other hand, two mutually exclusive constraints are presented
in ref. tidal1 , which proposed a limit between 70 $<~{}\Lambda_{1.4}~{}<$
580, and the PREX2 inferred values, whose limit lies between 642
$<~{}\Lambda_{1.4}~{}<$ 955 (PREX2, ). As hyperons are not present at a 1.4
$M_{\odot}$ star, we always have $R_{1}.4$ = 12.82 km and $\Lambda_{1.4}$ =
516 for the eL3$\omega\rho$, and $R_{1.4}$ = 13.68 km and $\Lambda_{1.4}$ =
696 for the GM1. Other results are presented in Tab. 2.

| $M_{max}/M_{\odot}$ | Hyp. at (fm-3) |  $R_{2.0}$ (km)
---|---|---|---
eL3$\omega\rho$ | 2.202 |  $\Lambda$ at 0.4114 | 12.379
eL3$\omega\rho$* | 2.206 |  $\Lambda$ at 0.4416 | 12.420
GM1 | 2.223 |  $\Lambda$ at 0.3264 | 13.092
GM1* | 2.208 |  $\Xi^{-}$ at 0.3655 | 13.193
Table 2:  Some of the neutron stars properties. Results with (without) *
indicate the presence (absence) of the $g_{\Sigma^{0}\Lambda\rho}$ coupling.

As can be seen, for massive neutron stars the influence of the
$g_{\Sigma^{0}\Lambda\rho}$ coupling is very limited. The
$g_{\Sigma^{0}\Lambda\rho}$ coupling causes a small increase of the maximum
mass, as well causes an increase of the radius for a fixed mass value. All
these increments are about only 0.5$\%$. This may sound a little disappointing
but we must remember that no one could know how strong would be the influence
of the $g_{\Sigma^{0}\Lambda\rho}$ until someone calculated its value.

---
Figure 3: (Color online) The EoS and the $v_{s}^{2}$ for neutron-$\Lambda$
matter. The dotted (solid) lines indicate the presence (absence) of the
$g_{\Sigma^{0}\Lambda\rho}$ coupling.

The effect of the $g_{\Sigma^{0}\omega\rho}$ coupling is more evident when we
consider a matter consisting of only neutrons and $\Lambda$’s. In ref. Gul2012
; James2016  the authors study a liquid-gas-like phase transition within
neutron-$\Lambda$ matter. The neutron-$\Lambda$ matter was also used to study
spinodal instability in ref. James2017 . Moreover, the existence of a neutral
bound state consisting of only neutrons and $\Lambda$’s was investigated in
ref. Rappold2013 ; Garcilazo2017 . Here I follow ref. James2017  and use
$\mu_{n}=\mu_{\Lambda}$. The EoS and the square of the speed of sound
$v_{s}^{2}=\partial p/\partial\epsilon$ are displayed in Fig. 3.

As can be seen, the presence of the $g_{\Sigma^{0}\Lambda\rho}$ stiffens the
EoS, as well as increases the speed of sound at high densities and pushes away
the onset of the $\Lambda$. For the eL3$\omega\rho$ the $\Lambda$ threshold is
pushed from 0.3634 fm-3 to 0.4164 fm-3, while within the GM1 parametrization
the onset is pushed from 0.2819 fm-3 to 0.3586 fm-3. For the GM1 the increase
of the density threshold is higher than 25$\%$, while for the eL3$\omega\rho$
it is around $15\%$.

Before I finish, I would like to mention that the applications of the
$g_{\Sigma^{0}\Lambda\rho}$ are far beyond those presented in this work. For
instance, it can potentially affect hypernuclei Aka2000  energy levels, as
well as hyperon-baryon scattering Rijken2010 .

##  4 Conclusions

In this work, I investigate the use of the symmetry groups and the SU(3)
Clebsch-Gordan coefficients to fix the coupling constants of the baryon octet
with the vector meson in order to keep the Yukawa Lagrangian as a singlet for
both: the antisymmetric and symmetric couplings. The main results of the
present work are summarized below:

  * •

I found that the current set of coupling constants for the SU(3) symmetry
group does not satisfy the relations of completeness and closure for the
symmetric coupling, while was already complete for the antisymmetric one
($\alpha_{V}$ = 1).

  * •

There are two additional Yukawa interactions related to the exchange of the
neutral $\rho$ meson between the $\Sigma^{0}$ and the $\Lambda$ hyperon. When
these interactions are taken into account the relations of completeness and
closure are restored.

  * •

Then I calculate the $g_{\Sigma^{0}\Lambda\rho}$ coupling constants within
SU(3) and SU(6) symmetry groups. In SU(6) we have $g_{\Sigma^{0}\Lambda\rho}$
= 0, and Sakurai’s theory of strong interaction is restored SAKURAI1960 .
Therefore, for the pure $F$-mode ($\alpha_{V}$ = 1) the
$g_{\Sigma^{0}\Lambda\rho}$ is not required to satisfy the SU(3) symmetry
group. However, if $\alpha_{V}~{}\neq~{}1$, the
$g_{\Sigma\Lambda\rho}~{}\neq~{}0$ and these interactions must be considered
to account for the completeness of the theory. These results are fully model-
independent.

  * •

The $\Lambda$-$\Sigma$ interaction is supported by experimental data, in the
so-called coherent $\Lambda-\Sigma$ coupling Aka2000 ; Aka2002

In order to study the effects of the $g_{\Sigma^{0}\Lambda\rho}$ couplings, I
add these crossed Yukawa couplings to the QHD model to study dense nuclear
matter, on which $\alpha_{V}$ is usually a free parameter.

  * •

I show that these crossed terms enter as off-diagonal terms in the
Hamiltonian. As a consequence, the coupling with the $\Lambda$ and with the
$\Sigma^{0}$ present opposite signs, despite having the same Clebsch-Gordan
coefficients.

  * •

I then obtain some numerical results for dense nuclear matter within two
different parametrizations: the eL$\omega\rho$ lopesPRD  and the GM1 GM1 . I
show that the $g_{\Sigma^{0}\Lambda\rho}$ coupling suppresses the $\Lambda$
onset whilst favoring the $\Xi^{-}$ one. In the case of the GM1, this is
enough to make the $\Xi^{-}$ the first hyperon to appear. In the case of
massive neutron stars, the $g_{\Sigma^{0}\Lambda\rho}$ coupling causes a very
small increase of the maximum masses and the radii for fixed masses (around
$0.5\%$).

  * •

Finally, I study a hadronic matter constituted by only neutrons and
$\Lambda$’s. I show that the $g_{\Sigma^{0}\Lambda\rho}$ coupling stiffens the
EoS, and pushes the hyperon threshold to higher densities. It also affects the
speed of the sound.

## Acknowledgements

The author was partially supported by CNPq Universal Grant No. 409029/2021-1.

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