# Neurosymbolic Meta-Reinforcement Lookahead Learning Achieves Safe Self-
Driving in Non-Stationary Environments

Haozhe Lei1 and Quanyan Zhu1 1The authors are with the Department of
Electrical and Computer Engineering, New York University, Brooklyn, NY, 11201,
USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the area of learning-driven artificial intelligence advancement, the
integration of machine learning (ML) into self-driving (SD) technology stands
as an impressive engineering feat. Yet, in real-world applications outside the
confines of controlled laboratory scenarios, the deployment of self-driving
technology assumes a life-critical role, necessitating heightened attention
from researchers towards both safety and efficiency. To illustrate, when a
self-driving model encounters an unfamiliar environment in real-time
execution, the focus must not solely revolve around enhancing its anticipated
performance; equal consideration must be given to ensuring its execution or
real-time adaptation maintains a requisite level of safety. This study
introduces an algorithm for online meta-reinforcement learning, employing
lookahead symbolic constraints based on _Neurosymbolic Meta-Reinforcement
Lookahead Learning_ (NUMERLA). NUMERLA proposes a lookahead updating mechanism
that harmonizes the efficiency of online adaptations with the overarching goal
of ensuring long-term safety. Experimental results demonstrate NUMERLA confers
the self-driving agent with the capacity for real-time adaptability, leading
to safe and self-adaptive driving under non-stationary urban human-vehicle
interaction scenarios.

###### Index Terms:

reinforcement learning, meta-learning, cyber security, autonomous vehicles,
human safety

##  I Introduction

The application of machine learning (ML) in self-driving (SD) technology
represents a marvel of engineering, enabling vehicles to process an array of
sensor inputs in real-time, interpret complex surroundings, and execute
actions with a precision that was once relegated to the realm of science
fiction. Recent advances in the field of machine learning, as evidenced by
works such as [1, 2, 3], have triggered a significant surge of curiosity and
investigation into the realm of learning-driven SD [4]. This application has
arisen in vehicles that can correctly work through known cityscapes,
anticipate pedestrian behavior, and interact perfectly with other vehicles,
all while following traffic rules and optimizing fuel efficiency.
Nevertheless, beyond controlled experimental setups, the inherent
unpredictability of artificial intelligence (AI) becomes evident when a self-
driving vehicle confronts a new and unfamiliar situation. In such instances,
the system’s performance might deteriorate or lead to a crash when
encountering unanticipated scenarios on real-world roads. The inaugural
instance of a pedestrian fatality attributed to autonomous vehicles surfaced
in 2018, when a self-driving Uber vehicle collided with a pedestrian crossing
an intersection in Tempe, Arizona, during the nighttime [5]. This tragic event
highlights the critical importance of improving the safety and adaptability of
autonomous driving systems. Considering such challenges, a pertinent question
arises: How can advances in technologies and methodologies help enhance the
capability of autonomous vehicles to operate safely and reliably in diverse
and complex environments?

Figure 1: An illustration of Neurosymbolic Meta-Reinforcement Lookahead
Learning. When driving in a changing environment, the agent first uses
observation from the environment to calibrate its belief at every time step
about the mode. Based on its belief, the agent conjectures its performance in
the future within a lookahead horizon. Then, using this conjecture, the agent
searches in its knowledge to find suitable safety constraints. In the
meantime, the knowledge of the agent will update itself by symbolic safety
constraint adaptation if needed. The policy is adapted through conjectural
lookahead optimization with safety constraints, leading to a suboptimal
(empirically) online control with a long-term safety guarantee.

·

In reinforcement learning (RL), stochastic policies are commonly employed in
partially observable environments, such as robotics or autonomous driving (no
matter in Q-learning methods [2, 6] or policy-based methods [7, 8]), where
agents encounter sensor noise and incomplete information. During offline
training, stochastic policies offer various advantages, including increased
robustness against uncertainty and environmental variations, improved
exploration capabilities, and compatibility with policy search algorithms like
evolutionary strategies or Monte Carlo (MC) methods [9]. However, the limited
generalization ability prevents RL from wide application in real SD systems
when encountering nonstationary environments different from their training
time. This drawback also makes the stochastic policies even more unstable in
the life-critical execution.

Enhancing the adaptability of reinforcement learning (RL) policies comprises
the objective of meta-reinforcement learning (meta-RL), which attempts to
discover a meta-policy capable of adeptly adjusting and delivering
satisfactory performance across a spectrum of environments [10]. While prior
works [10, 11, 12, 13] have dedicated significant efforts to this pursuit,
many of them continue to rely on offline methodologies. These approaches
display the capacity to adapt to a diverse array of tasks within environments
they are exposed to during their training in offline settings. The practical
implementation of online machine learning often faces challenges due to its
time-sensitive nature. Processing and updating models in real-time can be
demanding, making time a critical factor when deploying such systems in real-
world applications. Beyond the time constraints, another significant challenge
arises: the assurance of policy processing safety remains an ongoing concern
in these methods.

In conjunction with the real-time adaptation capability, several researchers
have incorporated safety-centric learning to support policy robustness. For
instance, in [14], an approach leveraging symbolism is proposed to formulate
distinct safety policies tailored to various state partitions. Similarly, [15]
presents a method that constructs a shield for actions based on observation
inputs, ensuring the safety of each individual step. Notably, neither of these
approaches accounts for the dynamic nature of the environment. This implies
that in scenarios where the initial environmental observations are incomplete
or where the environment is subject to change, the effectiveness of the safety
mechanisms might diminish.

Our Contributions In response to the dual challenges of limited real-time
adaptation capabilities and the quest for safety assurance, this study
introduces an algorithm for online meta-reinforcement learning, employing
lookahead symbolic constraints based on _Neurosymbolic Meta-Reinforcement
Lookahead Learning_ (NUMERLA). The underlying principle of NUMERLA is to
facilitate secure real-time learning by continually updating safety
constraints. The core idea involves employing logical statements as safety
constraints for the process of secure online meta-adaptation learning (OMAL)
[See Section II-B]. These constraints are iteratively refined in a forward-
looking manner during the online execution [See Equation NUMERLA]. This
lookahead updating mechanism balances the efficiency of online adaptations
with the overarching goal of ensuring long-term safety.

In summary, the main contributions of this work include: 1) conceptualizing
the challenge of acquiring adaptive strategies in a dynamic environment
characterized by symbolic safety constraints; 2) introducing an ensure-safe
Real-Time OMAL algorithm, which builds upon the principles of Neurosymbolic
Meta-Reinforcement Lookahead Learning (NUMERLA); 3) experimental results
demonstrating that NUMERLA enables the self-driving agent with the capacity
for real-time adaptability, leading to safe and self-adaptive driving under
non-stationary urban human-vehicle interaction scenarios.

##  II Definition and Model Structure of NUMERLA

###  II-A Meta Reinforcement learning

RL is a field that focuses on solving problems within a stationary environment
called a Markov Decision Process (MDP). Considering $z_{t}\in\mathcal{Z}$ is
the environment mode or latent variable hidden from the agent at time $t$. Let
$s_{t}\in\mathcal{S}$ and $a_{t}\in\mathcal{A}$ be the state input and the
control action at time $t$. In the context of RL, we often encounter
situations where the underlying conditions remain stable throughout a
decision-making period, known as $H$. This means that the parameters that
define the environment remain unchanged as time progresses (i.e., $z_{t}=z$).
This characteristic is known as ”stationarity,” which allows us to consider a
specific class of policies known as Markov policies [16]. These policies,
denoted as $\pi:\mathcal{S}\rightarrow\Delta(A)$, depend only on the current
state.

Denote a neural network-based policy by $\pi(s,a;\theta)$, where
$\theta\in\Theta\subset\mathbb{R}^{d}$ ($d$ represents the dimension of
parameters in the neural network). This choice aligns well with scenarios
where state information is collected from sensors. And the evaluation of the
choices will be given by reward function $r_{t}=r(s_{t},a_{t})$. The aim of RL
is to tackle a problem in the realm of stationary MDPs, where the environment
determines the best policy that maximizes the expectation of cumulative
rewards in a fixed environment $z$. These rewards are accounted for over time
using a discount factor $\gamma$ ($0<\gamma\leq 1$):

$\displaystyle\max_{\theta}J_{z}(\theta):=\mathbb{E}_{P(s_{t+1}|s_{t},a_{t};z),\pi(s_{t},a_{t};\theta)}\left[\sum_{t=1}^{H}\gamma^{t}r(s_{t},a_{t})\right].$
(RL)

Here, the transition $P(s_{t+1}|s_{t},a_{t};z_{t})$ tells how likely the agent
is to observe a certain state $s_{t+1}$ with control action $a_{t}$ under the
current mode conditions $z_{t}$. Since we are using the fixed $z_{t}=z$, it
can be reduced to $P(s_{t+1}|s_{t},a_{t};z)$.

Traditional meta-RL is explored in works like [10, 12, 13]. These methods aim
to discover a meta policy denoted as $\theta$, along with an adaptation
mapping called $\Phi$. The objective is to attain favorable rewards across
various environments by updating the meta policy $\theta$ through $\Phi$
within each environment. In simpler terms, they try to train a good meta
policy using offline methods. Instead of treating meta-learning as a fixed
optimization problem, we propose learning the meta-adaptation process in real-
time. This implies that the agent adjusts its adaptation strategies
continuously based on its observations. Essentially, our approach enables the
agent to adapt to changing environments. The following paragraph formally
defines the problem of online meta-adaptation learning (OMAL).

Let $\mathcal{I}_{t}=\\{s_{t},a_{t-1},r_{t-1}\\}$ be the set of agent’s
observations at time $t$, referred to as the information structure [17]. The
online adaptation mapping relies on the online observations
$\cup_{k=1}^{t}\mathcal{I}_{k}$. Then the meta adaptation mapping at time $t$
is defined as $\Phi_{t}(\theta):=\Phi(\theta,\cup_{k=1}^{t}\mathcal{I}_{k})$.
The adaptation mapping $\Phi$ adapts the meta policy $\theta$ to a new policy
fine-tuned for the specific $z$ at time step $t$ based on the agent’s
observations $\mathcal{I}_{t}$. Under this circumstance, we will use expected
reward
$r_{t}^{\pi}(s_{t};\theta):=\mathbb{E}_{a\sim\pi(\cdot|s_{t};\theta)}[r_{t}(s_{t},a)]$
as our new objective in the function shown below:

$\displaystyle\max_{\\{\Phi_{t}\\}_{t=1}^{H}}\quad\mathbb{E}_{z_{1},z_{2},\cdots,z_{H}}[\sum_{t=1}^{H}r^{\pi}(s_{t};\Phi_{t}(\theta))],$
(OMAL) s.t. $\displaystyle\qquad\qquad z_{t+1}\sim p_{z}(\cdot|z_{t}),\
t=1,\ldots,H-1,$
$\displaystyle\qquad\qquad\theta=\operatorname*{arg\,max}\mathbb{E}_{z\sim\rho_{z}}[J_{z}(\theta)].$

In this context, the mode denoted as $z\in\mathcal{Z}$ represents the specific
environment in which the offline policy is situated. Furthermore, we denote
the latent mode transitions probabilistically via a Markov chain
$p_{z}(z_{t+1}|z_{t})$ with an initial distribution denoted as
$\rho_{z}(z_{1})$. Our proposition involves the adoption of the Conjectural
Online Lookahead Adaptation (COLA) model, as outlined in [18] and expounded
upon in Section III-A, as a means to identify a viable $\Phi_{t}$. The diagram
illustration can also be found in Figure 1.

Figure 2: The NUMERLA framework is shown in the plot. Utilizing symbolic
logic-based safety constraints and online meta-adaptation learning techniques,
we consider the following scenario: With the state denoted as $s_{t}$, policy
as $\pi_{t}$, and policy constraint as $f_{i}$ at time $t$, generated by the
SSC function. The policy $\pi_{t}$ initiates dynamic adjustments online while
guided by the knowledge encoded in $f_{i}$. This adaptation process draws upon
insights from both the current state $s_{t}$ and historical context.
Subsequently, the revised policy $s_{t+1}$ governs the selection of an action,
thereby leading to the transition from state $s_{t}$ to $s_{t+1}$. Assuming
the environment mode space $\mathcal{Z}_{t}$ changes exclusively during steps
$1,4,6$. In case of such mode changes, the knowledge content is updated to
$f_{i+1}$.

###  II-B Objective function of OMAL with an action constraint

In the last section, we address the OMAL problem that can be solved by COLA,
which can be thought of as Neuro Lookahead Learning. This section explains how
symbolism can make the policy safer. When the OMAL wants to find an optimal
model that maximizes the online performance, the aim of NUMERLA is to further
make sure $K$ steps safety of the policy. The objective function is given by:

$\displaystyle\max_{\\{\Phi_{t}\\}_{t=1}^{H}}\quad\mathbb{E}_{z_{1},z_{2},\cdots,z_{H}}[\sum_{t=1}^{H}r^{\pi}(s_{t};\Phi_{t}(\theta))],$
(NUMERLA) s.t. $\displaystyle\qquad\qquad z_{t+1}\sim p_{z}(\cdot|z_{t}),\
t=1,\ldots,H-1,$
$\displaystyle\qquad\qquad\theta=\operatorname*{arg\,max}\mathbb{E}_{z\sim\rho_{z}}[J_{z}(\theta)],$
$\displaystyle\qquad\qquad\Phi_{t}(\theta)\in f_{t}(z_{t}),$
$\displaystyle\qquad\qquad f_{t}(z_{t}):=\begin{cases}\varphi_{1}&\text{ if
}\chi_{1}(z_{t})\\\ \varphi_{2}&\text{ if
}\chi_{2}(z_{t})\wedge\neg\chi_{1}(z_{t})\\\ \cdots&\\\ \varphi_{n}&\text{ if
}\chi_{n}(z_{t})\wedge\left(\bigwedge_{1\leq
i<n}\neg\chi_{i}(z_{t})\right)\end{cases}.$

where symbolic safety constraints (SSC)
$f_{t}:\mathcal{Z}_{t}\Rightarrow\Theta_{t}$ is a function belonging to the
space $\mathcal{F}$. The function $f_{t}$ serves to associate a mode within
$\mathcal{Z}_{t}$ with a subset of $\Theta_{t}$. It is important to note that
when alterations occur in the mode space $\mathcal{Z}_{t}$ at step $t$,
corresponding adjustments are made to both the mapping function $f_{t}$ and
the policy space $\Theta_{t}$ in accordance with the change. We define
$\mathcal{X}:=\\{\chi_{1},\ldots,\chi_{n}\\}$ as a collection of symbolic
logic judgments (expressed through linear predicates), which serve to segment
the space of modes. For the sake of clarity, we represent the non-overlapping
partitions as $\\{g_{1},\cdots,g_{n}\\}$, denoted by for all
$i\in\\{1,\cdots,n\\}$, $z^{\prime}\in g_{i}\subseteq\mathbf{Z}_{t}$ is a set
of mode that satisfies $\mbox{if
}\chi_{i}(z^{\prime})\wedge\left(\bigwedge_{1\leq
j<i}\neg\chi_{j}(z^{\prime})\right)\mbox{is true}$;
$\\{\varphi_{1},\cdots,\varphi_{n}\\}\subseteq\Theta$ are the symbolic logic-
based safety constraints which are the coupling between the knowledge mode
space with the physical action space. They can be defined as subsets in
$\Theta$ that include the safest action choices according to the yield
environment mode $z_{t}$. The framework of NUMERLA is also shown in Figure 2.

##  III Methodology of Optimization

###  III-A Conjectural Online Lookahead Adaptation

Following the model in [18], let $b_{t}$ be the agent’s belief (normally, the
belief is a pre-defined prediction or conjecture of the future mode in the
environment) and $\theta$ still be our obtained policy defined in Equation
OMAL. We consider a $K$ step future that can be represented by trajectory
$\tau_{t}^{K}:=(s_{t},a_{t},\ldots,s_{t+K-1},a_{t+K-1},s_{t+K})$. Following,
the distribution of trajectory $\tau_{t}^{K}$ can be characterized as:

$\displaystyle q(\tau_{t}^{K};b_{t},\theta):=$
$\displaystyle\prod_{k=0}^{K-1}\pi(a_{t+k}|s_{t+k};\theta)\prod_{k=0}^{K-1}\left[\sum_{z\in\mathcal{Z}}b_{t}(z)\underbrace{P(s_{t+k+1}|s_{t+k},a_{t+k};z)}_{\text{unkown}}\right].$

Here, the transition of the environment $P$ is unknown.

The goal of this model is to maximize the forecast future performance:

$\displaystyle\max_{\theta^{\prime}\in\Theta}\mathbb{E}_{q(\tau_{t}^{K};b,\theta^{\prime})}\sum_{k=0}^{K-1}r(s_{t+k},a_{t+k})$
(1)

However, the agent cannot access the distribution
$q(\tau_{t}^{K};b,\theta^{\prime})$ during the online adaptation. Thus, cannot
use policy gradient methods to solve the optimization problem.

As the replacement, we use importance sampling to do the optimization by
reformulating the original problem (1) to the conjectural lookahead
optimization (CLO) problem:

$\displaystyle\max_{\theta^{\prime}\in\Theta}$ $\displaystyle\
\mathbb{E}_{q(\cdot;b_{t},\theta)}\left[\prod_{k=0}^{K-1}\frac{\pi(a_{t+k}|s_{t+k};\theta^{\prime})}{\pi(a_{t+k}|s_{t+k};\theta)}\sum_{k=0}^{K-1}r(s_{t+k},a_{t+k})\right]$
(CLO) $\displaystyle\text{s.t. }\quad\mathbb{E}_{s\sim
q}D_{KL}(\pi(\cdot|s;\theta),\pi(\cdot|s;\theta^{\prime}))\leq\delta,$

where $D_{KL}$ is the Kullback-Leibler divergence. In the KL divergence
constraint, we slightly abuse the notation $q(\cdot)$ to denote the discounted
state visiting frequency $s\sim q$.

Equation CLO is equivalent to Equation 1 since the distribution difference
between $q(\tau_{t}^{K};b,\theta^{\prime})$ and $q(\tau_{t}^{K};b,\theta)$ in
(CLO) is compensated by the ratio
$\prod_{k=0}^{K-1}\frac{\pi(a_{t+k}|s_{t+k};\theta^{\prime})}{\pi(a_{t+k}|s_{t+k};\theta)}$.
When $\theta^{\prime}$ is close to the based policy $\theta$ in terms of KL
divergence, we can use the data collected during the training to finish the
approximation of the results. In the COLA setting, the data is gradient
sampling of the objective function in different environment modes. The overall
online updating process for the COLA is shown in Algorithm 1.

Algorithm 1 Conjectural Online Lookahead Adaptation

Input The meta policy $\theta$, belief $b$, training samples
$\\{\mathcal{D}_{z}\\}$, sample batch size $M$, lookahead horizon $K$

for $t\in\\{1,2,\ldots,\\}$ do

Acquire the sensor input $s_{t}$;

Implement the action using $\pi(\cdot|s_{t};\theta_{t})$;

Update the belief $b(z;s_{t})$;

Sample $M$ trajectories ($K$ steps from $t$ ) $\hat{\tau}_{t}^{K}$ under ${z}$
from $\\{\mathcal{D}_{z}\\}$;

Obtain $\theta^{\prime}$ by solving Conjecture Lookahead Optimization (CLO);

$\theta_{t+1}=\theta^{\prime}$.

###  III-B Symbolic Safety Constraint Adaptation

Figure 3: The evolution of the SSC function takes place through the absorption
of new information. Suppose the initial SSC function is $f_{0}$. We assume for
time step $1$ to $k$, $f_{0}$ can dominate everything. At $t=1$, the SSC uses
$\varphi_{n}$ as its constraint since $z_{1}\in g_{n}$. The lookahead
procedure conjectures the next time step should be in mode $z_{2}\in g_{1}$,
so the SSC will prepare to use $\varphi_{1}$ as the next constraint. The
knowledge update of SSC occurs when a novel mode is identified at $t=k$,
denoted as $z_{k+1}\notin g_{i}$ for all existing modes $g_{i}$ within the set
$\\{g_{1},\ldots,g_{n}\\}$, or in other words,
$z_{k+1}\notin\mbox{dom}(f_{0})$. This update can be executed through two
distinct approaches: either by integrating the new mode with an existing
earlier mode (solving Equation SSCAP with $g_{i},\mbox{ }\forall
i\in\\{1,\cdots,n\\}$) or by establishing a fresh partition exclusively for
the new mode (solving Equation SSCAP with $g_{n+1}$).

Denote a given safety assessment function
$Safe(s_{t},a_{t}):S\times\mathcal{A}\rightarrow\\{0,1\\}$ that outputs a
Boolean value where if state-action pair (s-a pair) $(s_{t},a_{t})$ is safe
(output 0) or unsafe (output 1). Then, for the symbolic safety constraint
adaptation (SSCA), its objective function can be defined as Equation SSCA
shown below:

$\displaystyle\min_{f}\sum_{z\in\mathcal{Z}}\sum_{\theta^{\prime}\in
f_{t}(b_{t}(z))}\mathbb{E}_{q(\tau_{t}^{K};b,\theta^{\prime})}\left[\sum_{k=0}^{K-1}Safe(s_{t+k},a_{t+k})\right].$
(SSCA)

On the other hand, we can divide it into optimization problems according to
different mode partitions $g_{i}$, called the symbolic safety constraint
adaptation of partition (SSCAP). Suppose we have:

$\displaystyle\hat{q}(\tau_{t}^{K};b_{t},g_{i},\theta):=$
$\displaystyle\prod_{k=0}^{K-1}\pi(a_{t+k}|s_{t+k};\theta)\prod_{k=0}^{K-1}\left[\sum_{z\in
g_{i}}b_{t}(z)P(s_{t+k+1}|s_{t+k},a_{t+k};z)\right].$

Then, we can denote the SSC optimization for specific partition:

$\displaystyle\min_{\varphi_{i}}\sum_{\theta^{\prime}\in\varphi_{i}}\mathbb{E}_{\hat{q}(\tau_{t}^{K};b_{t},g_{i},\theta)}\left[\sum_{k=0}^{K-1}Safe(s_{t+k},a_{t+k})\right].$
(SSCAP)

The foundational SSC function $f_{0}$ is derived through a heuristic process
rooted in prior human insights within our conceptual framework. It is
important to note that $\mathcal{Z}$ represents the range of modes entirely
encompassed by $f_{0}$. However, in scenarios where the agent is confronted
with a novel mode space denoted as $\mathcal{Z}^{\prime}\supset\mathcal{Z}$
demanding a more powerful SSC function, a knowledge expansion is imperative.
This expansion pertains to the enhancement of our understanding, specifically
the SSC function, to effectively accommodate this broader mode space. The
online update of the SCC can follow the rules described in Algorithm 2.

Algorithm 2 Symbolic Safety Constrain Adaptation

1:Input
$\\{\chi_{1},\ldots,\chi_{n}\\},\\{\varphi_{1},\cdots,\varphi_{n}\\},\mathcal{Z}^{\prime}$

2:Create partitions $\\{g_{1},\ldots,g_{n}\\}$ using
$\\{\chi_{1},\ldots,\chi_{n}\\}$

3:$g_{n+1}\leftarrow\emptyset$

4:for $z^{\prime}\in\mathcal{Z}^{\prime}$ do

5: if $z^{\prime}\notin g_{i},\forall g_{i}\in\\{g_{1},\ldots,g_{n}\\}$ then

6: $g_{n+1}\leftarrow g_{n+1}\cup\\{z^{\prime}\\}$

7:Find $g_{n+1}=\chi_{n+1}(z_{t})\wedge\left(\bigwedge_{1\leq
i<n+1}\neg\chi_{i}(z_{t})\right)$

8:Obtain $\varphi_{n+1}$ that optimal (SSCAP) with input $p_{n+1}$

9:Derive updated judgments $\\{\chi_{1},\ldots,\chi_{n+1}\\}$

10:return $\\{\chi_{1},\ldots,\chi_{n},\chi_{n+1}\\}$ and
$\\{\varphi_{1},\cdots,\varphi_{n},\varphi_{n+1}\\}$

Figure 3 shows the process of online updating of the SSC function. By
combining the results derived from Algorithm 2, we acquire the refined SSC
function denoted as $f_{1}$. It is essential to note that the enhancement of
the SSC function is not a solitary, instantaneous modification; rather, the
agent is required to gather data from the changing environment
$\mathcal{Z}^{\prime}$. This necessitates conducting multiple samplings from
the environment to achieve the desired refinement.

Illustrative examples can shed light on the process of updating the SCC. A
relevant exasmple is motivated by the disparities in driving practices across
different regions within the United States. Imagine a driver who has been
accustomed to the driving conditions in New York City but relocates to Texas.
This relocation exposes the driver to a distinct environmental context. In
urban traffic settings, the driver’s existing knowledge might still prove
effective. However, driving in Texas introduces new scenarios, such as
encountering wildlife like deer or bears on the road. Here, the driver not
only adapts through personal experience but can also seek insights from local
residents, or the acquisition of new modes online. Regarding the incorporation
of these novel modes into the driver’s cognitive framework, namely the
expansion of SCC, this can be accomplished by making minor adjustments to an
existing safety partition or creating a new partition catering exclusively to
these new modes. These concepts are illustrated in Figure 3.

In Section IV, our focus is only on the scenario where the SCC function
remains invariant throughout.

##  IV Experimental Configuration

For our experimental assessments, we employ CARLA-0.9.4 [19], a well-
established platform for urban self-driving scenarios. To establish the
communication between learning algorithms and environments, we adapt the API
by integrating the Multi-Agent Connected Autonomous Driving (MACAD) Gym [20]
framework atop CARLA.

We examine vehicle-human interactions in an urban traffic environment
featuring two agents: a vehicle with an initial velocity and a pedestrian,
illustrated in Figure 4. We denote the vehicle by $c$ and the pedestrian $p$.
To assess the effectiveness of our approach, we conduct experiments within two
distinct scenarios: one involving Well-Behaved walking and the other involving
jaywalking. Each scenario comprises three tasks determined by the initial
distance between the vehicle’s and pedestrian’s origin points. We will
describe more specific details later.

Figure 4: An illustration of the uncertain position of signal light scenario.
In this, we create a pedestrian with a signal light in front of the car on the
urban sidewalk road. The location of this pedestrian is uncertain. The sensors
will observe the velocities $v_{c,t},v_{p,t}$ and their distances to the
destination $d_{c,t},d_{p,t}$ of the pedestrian and the vehicle and the signal
light’s status $l_{t}$. The vehicle needs to reach its destination in a short
period of time without colliding with pedestrians.

·

We assume the state input is coming from the sensors on the vehicle. The state
representation comprises each agent’s current and previous speeds, denoted as
$v_{c,t},v_{p,t}\in\mathbb{R}$, and their distances to their respective
endpoints, represented by $d_{c,t},d_{p,t}\in\mathbb{R}$. Additionally, the
actions $a_{c,t}\in\mathcal{A}_{c}\subseteq\mathbb{R}^{n}$ and
$a_{p,t}\in\mathcal{A}_{p}\subseteq\mathbb{R}^{n}$ are included. Furthermore,
we introduce a simulated signal light input denoted as $l_{t}$, which serves
as an additional component within the state. It is important to highlight
that, since the sensors are only equipped on the vehicle, inputs stemming from
pedestrians and the signal light are initialized to $-1$ until the vehicle
approaches within a distance of $15$ meters from them. The complete structure
of the state $s_{t}$ encompasses 10 different variables, namely
$\\{d_{c,t},d_{p,t},v_{c,t},v_{p,t},l_{t},d_{c,t-1},d_{p,t-1},v_{c,t-1},v_{p,t-1},l_{t-1}\\}$.
When executing the SSC function $f_{t}$, we focus solely on the current state
information, represented as
$\hat{s}_{t}=\\{d_{ct},d_{pt},v_{ct},v_{pt},l_{t}\\}$, to ensure computational
efficiency.

For pedestrians and the vehicle agent, the available actions are defined in
Table I. In the case of pedestrians, the action values correspond to
acceleration towards the main road direction (if positive) or the opposite
direction (if negative). For the vehicle, the action values represent the
throttle strength (if positive) or the brake strength (if negative).

TABLE I: Discrete Actions # | Action | # | Action
---|---|---|---
0 | 0.0 | 4 | -1.0
1 | 1.0 | 5 | -0.5
2 | 0.5 | 6 | -0.25
3 | 0.25 |  |

The vehicle’s reward function hinges on its present velocity, proximity to the
destination, and the occurrence of collisions. Across every scenario,
encompassing both Well-Behaved walking and jaywalking scenarios, we sketch
three distinct initial gaps (15 meters, 25 meters, and 35 meters) between the
vehicle and pedestrians, classified according to their types. This methodology
serves to assess how effectively and safely the proposed NUMERLA model can
adeptly manage diverse traffic scenarios, thereby measuring its adaptability
and security.

In both scenarios, we assess the performance of the RL method, COLA method,
and NUMERLA method for each individual task. We capture the mean reward,
standard deviation, and collision rate as key metrics for each experimental
iteration.

###  IV-A Well-Behaved Walking

Figure 5: The performance comparison between RL, COLA, and NUMERLA for Well-
Behaved walking pedestrians where the value represents the mean rewards and
the error bar represents the standard deviation (std). The data is gathered
from 1,000 episodes of online executions. The RL performance is the worst of
the three types of methods, while the COLA obtains some better results.
However, both of these two methods return us to a poor std, which means
unstable performance. By using the NUMERLA method, we can achieve higher mean
rewards and a small std. It should be noted that the task 15-meter gets the
worst performance in every method. The reason is that the 15-meter is the
hardest task in this urban environment since our vehicle has an initial speed,
but the location of the pedestrians is too close.

In this scenario, the behavior of the pedestrian is guided by the signal
light. When the signal light is red, the pedestrian refrains from initiating
movement. When the signal light turns yellow, there is a 0.1 probability that
the pedestrian will commence walking. When the signal light switches to green,
the pedestrian promptly begins walking.

The Figure 5 shows the efficiency and long-term safety performance of the
NUMERLA method compared with the RL and the COLA in the Well-Behaved walking
scenario. We collect the collision rate, which means the ratio of episodes
with collision and the testing episode number, shown in Table II. We can find
collision rates are around zero for the NUMERLA method, which is much safer
than the other two methods.

Policy Type | Collision Rate
---|---
25m | 35m | 15m
RL | 0.251 | 0.303 | 0.718
COLA | 0.091 | 0.113 | 0.201
NUMERLA | 0.000 | 0.0004 | 0.000
TABLE II: Collision Rates for Well-Behaved Walking

###  IV-B Jaywalking

Figure 6: The performance comparison between RL, COLA, and NUMERLA for
jaywalking pedestrians where the value represents the mean rewards and the
error bar represents the std. The data is gathered from 1,000 episodes of
online executions. We should notice that all std get even bigger than the
experiments in Well-Behaved Walking since the pedestrians are unpredictable
under this circumstance. However, the performance of the reinforcement
learning (RL) technique excels in the 15-meter task compared to the compliant
walking scenario. This is attributed to pedestrians having the flexibility to
decide whether or not to cross the street, even opting for a significantly
delayed crossing if desired. But, our NUMERLA method still can maintain a high
mean reward and small std performance in this scenario.

In this scenario, the behavior of the pedestrian is unpredictable. The
pedestrian chooses a random time to initiate walking, irrespective of the
ongoing color of the signal light.

The Figure 6 shows the efficiency and long-term safety performance of the
NUMERLA method compared with the RL and the COLA, even though the pedestrians’
behavior is unpredictable in this scenario. In Table III, we can also find
collision rates are still around zero for the NUMERLA method, showing the
better performance of our method.

Policy Type | Collision Rate
---|---
25m | 35m | 15m
RL | 0.350 | 0.341 | 0.438
COLA | 0.156 | 0.154 | 0.190
NUMERLA | 0.000 | 0.0003 | 0.000
TABLE III: Collision Rates for Jaywalking

##  V Conclusion

This work has introduced a novel online meta-learning approach, building upon
the principles of Neurosymbolic Meta-Reinforcement Lookahead Learning
(NUMERLA). This technique guarantees the security of real-time learning by
consistently refining safety constraints. NUMERLA enables long-term safe
online adaptation by solving the conjectural lookahead optimization (CLO) and
the symbolic safety constraint adaptation (SSCA) on the fly using off-policy
data and the conjecture of the future.

## References

  * [1] S. Ross, G. Gordon, and D. Bagnell, “A Reduction of Imitation Learning and Structured Prediction to No-Regret Online Learning,” ser. Proceedings of Machine Learning Research, vol. 15. Fort Lauderdale, FL, USA: JMLR Workshop and Conference Proceedings, 2011, pp. 627—635. [Online]. Available: http://proceedings.mlr.press/v15/ross11a.html
  * [2] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” _Nature_ , vol. 518, no. 7540, pp. 529–533, 2015.
  * [3] T. Li, G. Peng, Q. Zhu, and T. Başar, “The confluence of networks, games, and learning a game-theoretic framework for multiagent decision making over networks,” _IEEE Control Systems Magazine_ , vol. 42, no. 4, pp. 35–67, 2022\.
  * [4] B. R. Kiran, I. Sobh, V. Talpaert, P. Mannion, A. A. A. Sallab, S. Yogamani, and P. Perez, “Deep Reinforcement Learning for Autonomous Driving: A Survey,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. PP, no. 99, pp. 1–18, 2021.
  * [5] D. Wakabayashi, “Self-driving uber car kills pedestrian in arizona, where robots roam,” _The New York Times_ , vol. 19, no. 03, 2018.
  * [6] T. Li and Q. Zhu, “On Convergence Rate of Adaptive Multiscale Value Function Approximation for Reinforcement Learning,” _2019 IEEE 29th International Workshop on Machine Learning for Signal Processing (MLSP)_ , pp. 1–6, 2019.
  * [7] R. S. Sutton, D. A. McAllester, S. P. Singh, and Y. Mansour, “Policy gradient methods for reinforcement learning with function approximation,” in _Advances in Neural Information Processing Systems 12_. MIT press, 2000, pp. 1057—1063. [Online]. Available: http://papers.nips.cc/paper/1713-policy-gradient-methods-for-reinforcement-learning-with-function-approximation.pdf
  * [8] T. Li, G. Peng, and Q. Zhu, “Blackwell Online Learning for Markov Decision Processes,” _2021 55th Annual Conference on Information Sciences and Systems (CISS)_ , vol. 00, pp. 1–6, 2021.
  * [9] D. Silver and J. Veness, “Monte-Carlo Planning in Large POMDPs,” in _Advances in Neural Information Processing Systems_ , vol. 23. Curran Associates, Inc.
  * [10] A. Nagabandi, I. Clavera, S. Liu, R. S. Fearing, P. Abbeel, S. Levine, and C. Finn, “Learning to Adapt in Dynamic, Real-World Environments Through Meta-Reinforcement Learning,” _arXiv_ , 2018.
  * [11] C. Finn, P. Abbeel, and S. Levine, “Model-agnostic meta-learning for fast adaptation of deep networks,” in _International Conference on Machine Learning_. PMLR, 2017, pp. 1126–1135.
  * [12] K. Rakelly, A. Zhou, C. Finn, S. Levine, and D. Quillen, “Efficient off-policy meta-reinforcement learning via probabilistic context variables,” in _Proceedings of the 36th International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, K. Chaudhuri and R. Salakhutdinov, Eds., vol. 97. PMLR, 09–15 Jun 2019, pp. 5331–5340. [Online]. Available: https://proceedings.mlr.press/v97/rakelly19a.html
  * [13] J. Harrison, A. Sharma, C. Finn, and M. Pavone, “Continuous Meta-Learning without Tasks,” in _Advances in Neural Information Processing Systems_ , vol. 33. Curran Associates, Inc., pp. 17 571–17 581.
  * [14] G. Anderson, A. Verma, I. Dillig, and S. Chaudhuri, “Neurosymbolic reinforcement learning with formally verified exploration,” 2020.
  * [15] M. Alshiekh, R. Bloem, R. Ehlers, B. Könighofer, S. Niekum, and U. Topcu, “Safe reinforcement learning via shielding,” 2017.
  * [16] M. L. Puterman, _Markov Decision Processes: Discrete Stochastic Dynamic Programming_ , 1st ed. USA: John Wiley & Sons, Inc., 1994.
  * [17] T. Li, Y. Zhao, and Q. Zhu, “The role of information structures in game-theoretic multi-agent learning,” _Annual Reviews in Control_ , vol. 53, pp. 296–314, 2022.
  * [18] T. Li, H. Lei, and Q. Zhu, “Self-adaptive driving in nonstationary environments through conjectural online lookahead adaptation,” in _2023 IEEE International Conference on Robotics and Automation (ICRA)_ , 2023, pp. 7205–7211.
  * [19] A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, and V. Koltun, “CARLA: An open urban driving simulator,” in _Proceedings of the 1st Annual Conference on Robot Learning_ , 2017, pp. 1–16.
  * [20] P. Palanisamy, “Multi-agent connected autonomous driving using deep reinforcement learning,” 2019.


# Analytical Insights on Hadronic Top Quark Polarimetry

Zhongtian Dong<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Kansas, Lawrence, KS 66045, USA  Dorival Gonçalves<EMAIL_ADDRESS>Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA
Kyoungchul Kong<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Kansas, Lawrence, KS 66045, USA  Andrew J. Larkoski<EMAIL_ADDRESS>Department of Physics and Astronomy, University of California, Los Angeles, CA
90095, USA
Mani L. Bhaumik Institute for Theoretical Physics, University of California,
Los Angeles, CA 90095, USA  Alberto Navarro
<EMAIL_ADDRESS>Department of Physics, Oklahoma State
University, Stillwater, OK, 74078, USA

###### Abstract

Top quark polarization provides an important tool for studying its production
mechanisms, spin correlations, top quark properties, and new physics searches.
Unlike lighter quarks, the top quark’s polarization remains intact until its
decay, enabling precise spin measurements. While the down-type fermions from
$W$ boson decay are known to be effective spin analyzers, charged leptons have
typically been the main target for most analyses. In this paper, we
investigate the relevance of global jet dynamics – considering kinematics, jet
charges, and particle multiplicity – for hadronic top quark polarimetry. The
formalism used allows for analytical derivations obtained throughout the
manuscript, offering deeper insights into the corresponding phenomenology.

##  I Introduction

Top quark polarization is crucial for understanding its production mechanisms
and shedding light into spin dynamics through observable decay patterns.
Unlike lighter quarks, the top quark decays before its polarization is
disrupted by soft QCD interactions, enabling precise spin measurements. Spin
correlations between top quark pairs reveal complex structures, enhancing our
understanding beyond event rates and making them useful in new physics
searches Bernreuther:1993hq ; Arai:2007ts ; Perelstein:2008zt ; Krohn:2011tw ;
Barger:2011pu ; Baumgart:2011wk ; Buckley:2015vsa ; Buckley:2015ctj ;
Goncalves:2016qhh ; Goncalves:2018agy ; Barger:2019ccj ; Goncalves:2021dcu ;
Barman:2021yfh ; MammenAbraham:2022yxp ; Maltoni:2024tul . Recent studies at
the Large Hadron Collider (LHC) not only confirm these correlations
Mahlon:1995zn ; Mahlon:2010gw ; CDF:2010yag ; D0:2011psp ; ATLAS:2019zrq ;
CMS:2019nrx ; Aguilar-Saavedra:2021ngj ; Aguilar-Saavedra:2022kgy , but also
demonstrate that top quarks can be entangled in certain kinematic regimes
Afik:2020onf ; Fabbrichesi:2021npl ; Severi:2021cnj ; Aguilar-Saavedra:2022uye
; Afik:2022kwm ; Afik:2022dgh ; Severi:2022qjy ; Aoude:2022imd ;
PhysRevD.109.115023 ; Han:2023fci ; Cheng:2023qmz ; ATLAS:2023fsd ;
Barr:2024djo ; CMS-PAS-TOP-23-001 ; CMS:2024vqh ; Dong:2024xsg ; Cheng:2024btk
, motivating further research with increasing energy and luminosity to explore
other quantum correlations that become more phenomenologicaly relevant at high
energy scales, such as Bell’s inequalities PhysRevD.109.115023 .

The down-type fermion from the $W$ boson decay is the most effective spin
analyzer for the top quark, providing particularly clean measurements in
leptonic decays Bernreuther:2010ny . However, there is significant interest in
measuring the polarization of top quarks that decay hadronically due to their
sizable branching ratio. Conventionally, it is assumed that up to 50% of the
spin analyzing power can be restored in such scenarios using the softest of
the two light jets from the top decay in the top quark rest frame
Jezabek:1994qs . Ref. Tweedie:2014yda  examined a straightforward method for
analyzing spin in hadronic decays by using top quark decay kinematics,
achieving a spin analyzing power of 64% at leading order. This has been
proposed as a promising kinematic method to probe entanglement and Bell’s
inequalities in the semileptonic channel of $t\bar{t}$ production at the LHC
PhysRevD.109.115023 ; Han:2023fci . More recently, we have proposed a jet
flavor tagging method and demonstrated the improvements to hadronic top quark
polarimetry on simulated data using a Graph Neural Network Dong:2024xsg . We
find that the spin analyzing power can be improved to approximately 0.75
(0.86), assuming an efficiency of 0.5 (0.2) for the network.

In this paper, we apply the general formalism to hadronic top quark
polarimetry presented in Ref. Dong:2024xsg  to investigate the relevance of
global jet dynamics – arising from kinematics, jet charges Field:1977fa ;
Krohn:2012fg ; Fraser:2018ieu ; Kang:2023ptt , and particle multiplicity – on
enhancing the sensitivity to hadronic top quark polarimetry. We aim to provide
an analytical understanding for this subset of observables used in our
previous machine learning study Dong:2024xsg . In Sec. II, we adopt an
analytical approach, providing parametric estimates of the improvement with
helicity angle, jet charge, and particle multiplicity. In Sec. III, we study
the phenomenological relevance of these observables, highlighting the phase
space where they lead to substantial improvements. We conclude in Sec. IV.
Appendix A is reserved for a simple proof demonstrating that incorporating jet
charge information always enhances the spin analyzing power.

##  II Theoretical framework

In this section, we explore the influence of helicity angle, jet charge, and
particle multiplicity on the spin analyzing power in hadronic top quark
decays, exploring the general formalism presented in Ref. Dong:2024xsg . We
begin by reviewing the fundamental definitions for top quark polarization and
jet charges. Following this, we apply the general formalism from Ref.
Dong:2024xsg  to the observables under consideration.

###  II.1 Top Quark Polarization

Due to its short lifetime, the polarization axis of the top quark $\vec{p}$,
with $0\leq|\vec{p\,}|\leq 1$, correlates with its decay products
Mahlon:2010gw ; ParticleDataGroup:2022pth  as

$\displaystyle\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta_{k}}=\frac{1}{2}\left(1+\beta_{k}p\cos\theta_{k}\right)\,,$
(1)

where $\cos\theta_{k}\equiv\hat{p}\cdot\hat{k}$ with $\hat{k}$ being the
direction of a particular final state particle $k$ (calculated in the top rest
frame), $p$ is the degree of polarization of the ensemble, $\beta_{k}$ is the
spin analyzing power of the final state $k$, and $\Gamma$ is the top quark
partial decay width. Among the top quark final states, the direction $\hat{d}$
of the down-type fermion from the $W$ decay displays maximal spin analyzing
power due to the $V-A$ structure of the weak interactions, leading to
$\beta_{\ell^{+},\bar{d}}=1$. Whereas the charged lepton from the top quark
decay can be uniquely identified, pinpointing the down-type quark from
hadronic top quark decay in a collider environment is more challenging.
Nevertheless, a proxy for the down-type quark direction can be defined by a
probability weighted vector using a combination of kinematics and jet flavor
tagging Tweedie:2014yda ; Dong:2024xsg . Generally, this direction can be
defined as a combination of the soft $\hat{q}_{\text{soft}}$ and hard
$\hat{q}_{\text{hard}}$ jet directions from the $W$ boson decay Dong:2024xsg :

$\displaystyle\vec{q}_{\text{opt}}=\,\,p(\bar{d}\to q_{\text{hard}}|\\{{\cal
O}\\})\,\hat{q}_{\text{hard}}+p(\bar{d}\to q_{\text{soft}}|\\{{\cal
O}\\})\,\hat{q}_{\text{soft}}\,,$ (2)

where $\\{\cal O\\}$ represents a set of observables, such as helicity angle,
jet charge, and particle multiplicity. The spin analyzing power for this
general direction is obtained from the vector’s length $\beta_{\rm
opt}\equiv|\vec{q}_{\rm opt}|$, where $0\leq|\vec{q}_{\rm opt}|\leq 1$. In
this article, we assume that the $b$-quark from the hadronic top decay has
already been identified.

###  II.2 Jet Charge

A definition of jet charge is given by Field:1977fa ; Kang:2023ptt

$\displaystyle{\cal Q}_{\kappa}\equiv\sum_{i\in J}\,z_{i}^{\kappa}Q_{i}\,,$
(3)

where the sum runs over all particles $i$ in the jet $J$, $Q_{i}$ is the
electric charge of particle $i$, $z_{i}$ is its the energy fraction of
particle $i$ in a given jet, and $\kappa>0$ is a parameter introduced to
ensure the infrared safety of the jet charge. Although the jet charge
distribution can be computed from data, under a few assumptions, the
functional form of the jet charge distribution conditioned on particle
multiplicity can be approximated with a Gaussian function with appropriate
mean and variance Kang:2023ptt . Therefore, we can estimate the impact of the
jet charge in the spin analyzing power analytically. We note that the average
charged particle multiplicity from on-shell $W$ decay has been measured to be
$\langle N_{\text{charged}}\rangle=19.39$, while the average charged pion
multiplicity is $\langle N_{\pi^{\pm}}\rangle=15.7$ ParticleDataGroup:2022pth
.

###  II.3 Conditional Probabilities and Spin Analyzing Power

To study the top quark polarimetry, we first determine the probability that
the harder of the two jets from the $W$ decay, in the rest frame of the top
quark, is initiated by a down-type quark. For concreteness, we will consider
the top quark decay $t\to b(W^{+}\to\bar{d}u)$ in the following discussion.
This probability is calculated based on the measured values of a set of
observables. In this paper, the observables we consider are:

  * •

the cosine of the helicity angle $c_{W}$ (the angle between the down-type
quark and opposite direction of the bottom quark in the frame of the $W$
boson),

  * •

the jet charges of the hard and soft jets, ${\cal Q}_{\kappa,h}$ and ${\cal
Q}_{\kappa,s}$, and

  * •

particle multiplicity in the two jets, $N_{h}$ and $N_{s}$.

We start by considering only measurements of the helicity angle and jet
charges, and work to simplify the relevant probability of Eq. (2) following
some reasonable assumptions. For the case at hand, we want to compute the
conditional probability

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})=\frac{p(\bar{d}\to q_{\text{hard}},c_{W},{\cal
Q}_{\kappa,h},{\cal Q}_{\kappa,s})}{p(c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})}\,.$ (4)

The key assumption we make is that the jet charge measurements on the two $W$
decay product jets are independent of one another, and so the denominator
factorizes to

$\displaystyle p(c_{W},{\cal Q}_{\kappa,h},{\cal Q}_{\kappa,s})$
$\displaystyle=p({\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s}|c_{W})\,p(c_{W})=p({\cal Q}_{\kappa,h}|c_{W})\,p({\cal
Q}_{\kappa,s}|c_{W})\,p(c_{W})\,.$ (5)

We believe this is a reasonable assumption because only the sum of the mean
values of the jet charges of the decay product jets is constrained to
reproduce the electric charge of the $W$ boson. The conditional probability
can then be written as

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})=\frac{p(\bar{d}\to q_{\text{hard}},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s}|c_{W})}{p({\cal Q}_{\kappa,h}|c_{W})\,p({\cal
Q}_{\kappa,s}|c_{W})}\,.$ (6)

Going further, we decompose the numerator

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})=\frac{p({\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s}|c_{W},\bar{d}\to q_{\text{hard}})\,p(\bar{d}\to
q_{\text{hard}}|c_{W})}{p({\cal Q}_{\kappa,h}|c_{W})\,p({\cal
Q}_{\kappa,s}|c_{W})}\,,$ (7)

where $p(\bar{d}\to q_{\text{hard}}|c_{W})$ is the probability identified to
define the optimal polarization axis using exclusively kinematic information
based on Ref. Tweedie:2014yda . Since the probability in the numerator also
factorizes assuming the independence of jet charge measurements, we have

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})=\frac{p({\cal Q}_{\kappa,h}|c_{W},\bar{d}\to
q_{\text{hard}})\,p({\cal Q}_{\kappa,s}|c_{W},\bar{d}\to
q_{\text{hard}})\,p(\bar{d}\to q_{\text{hard}}|c_{W})}{p({\cal
Q}_{\kappa,h}|c_{W})\,p({\cal Q}_{\kappa,s}|c_{W})}\,.$ (8)

With a fixed flavor for the jet, the jet charge measurement is independent of
kinematics, simplifying the numerator:

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})=\frac{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}})\,p({\cal Q}_{\kappa,s}|u\to q_{\text{soft}})\,p(\bar{d}\to
q_{\text{hard}}|c_{W})}{p({\cal Q}_{\kappa,h}|c_{W})\,p({\cal
Q}_{\kappa,s}|c_{W})}\,.$ (9)

Next, we establish the probability $p({\cal Q}_{\kappa,h}|c_{W})$, which can
be defined by summing over all possible hard jets using the conditional
probability

$\displaystyle p({\cal Q}_{\kappa,h}|c_{W})=p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}})\,p(\bar{d}\to q_{\text{hard}}|c_{W})+p({\cal
Q}_{\kappa,h}|u\to q_{\text{hard}})\,p(u\to q_{\text{hard}}|c_{W})\,,$ (10)

with a similar expression for $p({\cal Q}_{\kappa,s}|c_{W})$. Thus, the
conditional probability of interest is given by

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s})$ $\displaystyle=\frac{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}})}{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}})\,p(\bar{d}\to q_{\text{hard}}|c_{W})+p({\cal
Q}_{\kappa,h}|u\to q_{\text{hard}})\,p(u\to q_{\text{hard}}|c_{W})}$
$\displaystyle\times\frac{p({\cal Q}_{\kappa,s}|u\to q_{\text{soft}})}{p({\cal
Q}_{\kappa,s}|\bar{d}\to q_{\text{soft}})\,p(\bar{d}\to
q_{\text{soft}}|c_{W})+p({\cal Q}_{\kappa,s}|u\to q_{\text{soft}})\,p(u\to
q_{\text{soft}}|c_{W})}$ (11) $\displaystyle\times p(\bar{d}\to
q_{\text{hard}}|c_{W})\,.$

This result can be readily extended to include the measurements of particle
multiplicity in the two jets, where

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})=\frac{p({\cal
Q}_{\kappa,h},N_{h}|\bar{d}\to q_{\text{hard}})}{p({\cal
Q}_{\kappa,h},N_{h}|\bar{d}\to q_{\text{hard}})p(\bar{d}\to
q_{\text{hard}}|c_{W})+p({\cal Q}_{\kappa,h},N_{h}|u\to q_{\text{hard}})p(u\to
q_{\text{hard}}|c_{W})}$ $\displaystyle\hskip 153.6447pt\times\frac{p({\cal
Q}_{\kappa,s},N_{s}|u\to q_{\text{soft}})}{p({\cal
Q}_{\kappa,s},N_{s}|\bar{d}\to q_{\text{soft}})p(\bar{d}\to
q_{\text{soft}}|c_{W})+p({\cal Q}_{\kappa,s},N_{s}|u\to q_{\text{soft}})p(u\to
q_{\text{soft}}|c_{W})}$ $\displaystyle\hskip 153.6447pt\times p(\bar{d}\to
q_{\text{hard}}|c_{W})\,.$ (12)

If we assume that the quarks are first generation and isospin is an exact
symmetry, then the particle multiplicity distribution of the $\bar{d}$ versus
$u$ jets is identical, making the overall dependence on multiplicity
distribution cancel out. We then find

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})$ $\displaystyle=\frac{p({\cal
Q}_{\kappa,h}|\bar{d}\to q_{\text{hard}},N_{h})}{p({\cal
Q}_{\kappa,h}|\bar{d}\to q_{\text{hard}},N_{h})p(\bar{d}\to
q_{\text{hard}}|c_{W})+p({\cal Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})p(u\to
q_{\text{hard}}|c_{W})}$ $\displaystyle\times\frac{p({\cal Q}_{\kappa,s}|u\to
q_{\text{soft}},N_{s})}{p({\cal Q}_{\kappa,s}|\bar{d}\to
q_{\text{soft}},N_{s})p(\bar{d}\to q_{\text{soft}}|c_{W})+p({\cal
Q}_{\kappa,s}|u\to q_{\text{soft}},N_{s})p(u\to q_{\text{soft}}|c_{W})}$
$\displaystyle\times p(\bar{d}\to q_{\text{hard}}|c_{W})\,.$ (13)

The universality assumption in the particle multiplicity between first and
second generations is in general not accurate. However, given the limitations
of collider experiments, it serves as a reasonable approximation for practical
purposes. For instance, while we expect more kaons from the second generation
quarks, achieving high accuracy in discriminating between kaons and pions is
very challenging. This universality assumption provides a conservative
estimate on the conditional probability above, offering only minimal
enhancement to top quark polarimetry. Any additional information would
invariably lead to further improvements.111Discrimination between up-type and
down-type jets can be enhanced by incorporating trajectory data from the
tracking system, using charm tagging techniques ATLAS:2018mgv ; CMS-
DP-2023-006 . We leave these improvements for the second-generation fermions
from charm tagging for a future study.

Finally, we observe that only the likelihood ratio of the jet charge
distribution is necessary, where

$\displaystyle p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})$
$\displaystyle=\frac{1}{p(\bar{d}\to q_{\text{hard}}|c_{W})+\frac{p({\cal
Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})}{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}},N_{h})}\,p(u\to q_{\text{hard}}|c_{W})}$
$\displaystyle\times\frac{1}{\frac{p({\cal Q}_{\kappa,s}|\bar{d}\to
q_{\text{soft}},N_{s})}{p({\cal Q}_{\kappa,s}|u\to
q_{\text{soft}},N_{s})}\,p(\bar{d}\to q_{\text{soft}}|c_{W})+p(u\to
q_{\text{soft}}|c_{W})}$ (14) $\displaystyle\times p(\bar{d}\to
q_{\text{hard}}|c_{W})\,.$

Notice that the jet charge dependence appears in two factors: one for the
charge of the hard jet and another for the charge of the soft jet. The first
factor depends on the likelihood ratio of the $u$ quark leading to the hard
jet over the $\bar{d}$ quark resulting in the hard jet. For a fixed $N_{h}$,
the mean of $p({\cal Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})$ is slightly
more positive than the mean of $p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}},N_{h})$, see Eq. (II.3). Hence, for large positive values of
${\cal Q}_{\kappa,h}$ the likelihood ratio $\frac{p({\cal Q}_{\kappa,h}|u\to
q_{\text{hard}},N_{h})}{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}},N_{h})}$ becomes very large and the right-hand side of Eq.
(14) approaches zero. This suggests that the average vector length should show
an enhancement for large positive values of ${\cal Q}_{\kappa,h}$, resulting
in a large spin analyzing power $\beta_{opt}$, see Eq. (2). Using a similar
argument, we can conclude that the ratio $\frac{p({\cal
Q}_{\kappa,s}|\bar{d}\to q_{\text{soft}},N_{s})}{p({\cal Q}_{\kappa,s}|u\to
q_{\text{soft}},N_{s})}$ becomes sizable for large negative values of ${\cal
Q}_{\kappa,s}$, and we should expect an enhancement in spin analyzing power
$\beta_{opt}$ for this limit as well.

In the limit of a large multiplicity $N$ and assuming independent emissions,
the general form of the jet charge distribution conditioned on multiplicity is
Gaussian, which can be expressed as Kang:2023ptt

$\displaystyle p({\cal Q}_{\kappa}|\bar{d},N)$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-\frac{({\cal
Q}_{\kappa}-\mu_{\bar{d}})^{2}}{2\sigma^{2}}}\,,$ (15) $\displaystyle p({\cal
Q}_{\kappa}|u,N)$
$\displaystyle=\frac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-\frac{({\cal
Q}_{\kappa}-\mu_{u})^{2}}{2\sigma^{2}}}\,.$ (16)

Assuming that all particles in the jets are only pions and exact isospin
conservation holds (see comments on this assumption below Eq. (13)), we have

$\displaystyle\mu_{\bar{d}}$
$\displaystyle=\frac{1}{3}N^{-\kappa}\left(1+\frac{\kappa}{2}(\kappa-1)\sigma_{z}^{2}N^{2}+\cdots\right),$
$\displaystyle\mu_{u}$
$\displaystyle=\frac{2}{3}N^{-\kappa}\left(1+\frac{\kappa}{2}(\kappa-1)\sigma_{z}^{2}N^{2}+\cdots\right),$
(17) $\displaystyle\sigma^{2}$
$\displaystyle=\frac{2}{3}N^{1-2\kappa}\left(1+\kappa(2\kappa-1)\sigma_{z}^{2}N^{2}+\cdots\right)\,.$

Since the soft and hard jets can display different particle multiplicities,
$N_{s}$ and $N_{h}$, the means and variance would differ depending on whether
the up/down quarks are identified as the soft/hard jets. In general, this
implies that $\sigma_{z}^{2}N^{2}$ could vary for each of these cases. For
simplicity, we assume that this term is the same for all distributions.

These expressions explicitly depend on the energy weighting exponent $\kappa$
from the definition of the jet charge, see Eq. (3). We have included
contributions through the variance $\sigma_{z}^{2}$ of the multiplicity-
conditioned energy fraction distribution $p(z|N)$, where we note that

$\displaystyle\langle z\rangle$ $\displaystyle=\int
dz\,z\,p(z|N)=\frac{1}{N}\,,$ (18) $\displaystyle\sigma_{z}^{2}$
$\displaystyle=\int dz\,\left(z-\frac{1}{N}\right)^{2}\,p(z|N)\,.$ (19)

Although $\sigma_{z}^{2}$ is in principle unknown, it is strictly non-
negative, bounded, and can be assumed to scale as $\sigma_{z}^{2}\sim 1/N^{2}$
Kang:2023ptt ; Kang:2023zdx .

##  III Results

While the evaluation of the mean spin resolving power must be performed
numerically, the mean of the square resolving power can be expressed in closed
form. When using exclusively kinematic information, the square spin resolving
power is given by

$\displaystyle|{\vec{q}}_{\text{opt}}^{\text{ kin}}|^{2}=1-2\,p(\bar{d}\to
q_{\text{hard}}|c_{W})p(\bar{d}\to
q_{\text{soft}}|c_{W})\left(1-\hat{q}_{\text{hard}}\cdot\hat{q}_{\text{soft}}\right)\,.$
(20)

Here, the angle between the hard and soft jets in the top rest frame can be
expressed in terms of the helicity angle $\theta_{W}$ as

$\displaystyle
1-\hat{q}_{\text{hard}}\cdot\hat{q}_{\text{soft}}=\frac{2}{1+\frac{(m_{t}^{2}-m_{W}^{2})^{2}}{4m_{t}^{2}m_{W}^{2}}\,\sin^{2}\theta_{W}}\,,$
(21)

where $m_{t}$ and $m_{W}$ are the masses of the top quark and $W$ boson,
respectively. In this estimation, we neglect the effects of the $b$-quark
mass. The mean of the square resolving power is then obtained by integrating
$|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt kin}}|^{2}$ over the
distribution of the helicity angle, $p(c_{W})$,

$\displaystyle\langle|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt
kin}}|^{2}\rangle$
$\displaystyle=\int\,dc_{W}\,p(c_{W})\,|\vec{q}_{\text{opt}}^{\text{\hskip
0.19926pt kin}}|^{2}\,,$ (22)

where the helicity angle distribution $p(c_{W})$ is defined as Bern:2011ie

$\displaystyle p(c_{W})=\frac{3}{8}f_{R}(1\pm
c_{W})^{2}+\frac{3}{4}f_{0}(1-c_{W}^{2})+\frac{3}{8}f_{L}(1\mp c_{W})^{2}\,.$
(23)

The upper sign corresponds to $W^{+}$ and the lower sign to $W^{-}$. The
polarization fractions can be obtained from the following relations

$\displaystyle f_{0}=2-5\langle c_{W}^{2}\rangle,\hskip
28.45274ptf_{L}=-\frac{1}{2}\mp\langle c_{W}\rangle+\frac{5}{2}\langle
c_{W}^{2}\rangle,\hskip 28.45274ptf_{R}=-\frac{1}{2}\pm\langle
c_{W}\rangle+\frac{5}{2}\langle c_{W}^{2}\rangle\,,$ (24)

where the fractions for the zero helicity, left-handed helicity, and right-
handed helicity for the $W^{+}$ boson in the top quark rest frame are given by
$f_{0}\approx 0.7$, $f_{L}\approx 0.3$, and $f_{R}=0$, respectively. For the
$W^{-}$ decay, the quoted values for $f_{R}$ and $f_{L}$ are exchanged, as can
be inferred from Eq. (24).

The integral from Eq. (22) evaluates to

$\displaystyle\langle|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt
kin}}|^{2}\rangle$
$\displaystyle=\frac{m_{t}^{4}-5m_{t}^{2}m_{W}^{2}-2m_{W}^{4}}{m_{t}^{4}+m_{t}^{2}m_{W}^{2}-2m_{W}^{4}}+\frac{12m_{t}^{2}m_{W}^{4}}{(m_{t}^{4}+m_{t}^{2}m_{W}^{2}-2m_{W}^{4})\sqrt{m_{t}^{4}-m_{W}^{4}}}\left(\tanh^{-1}\frac{\sqrt{m_{t}^{2}-m_{W}^{2}}}{\sqrt{m_{t}^{2}+m_{W}^{2}}}\right)\,.$
(25)

Plugging in the values for $m_{t}=172.57\,$GeV and $m_{W}=80.3692\,$GeV
ParticleDataGroup:2022pth , we obtain

$\displaystyle\sqrt{\langle|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt
kin}}|^{2}\rangle}\approx 0.640\,,$ (26)

which is slightly larger than the mean spin resolving power
$\langle|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt kin}}|\rangle\approx
0.638$. Hence, $\sqrt{\langle|\vec{q}_{\text{opt}}^{\text{\hskip 0.19926pt
kin}}|^{2}\rangle}$ also works as a good proxy for $\beta_{\rm opt}$, and we
shall use it to numerically estimate the effect of the jet charge on the spin
analyzing power. This is done by generalizing Eq. (22) to account for the
average of $|\vec{q}_{\text{opt}}|^{2}$ over the charge of the hard and soft
jets, and over the (cosine of the) helicity angle for a fixed particle
multiplicity222In Appendix A, we prove, using these results and general
arguments, that the spin analyzing power is enhanced by incorporating jet
charge information.

$\displaystyle\langle|\vec{q}_{\text{opt}}|^{2}\rangle=\int\,dc_{W}\,p(c_{W})\int\,d{\cal
Q}_{\kappa,h}\,d{\cal Q}_{\kappa,s}\,p({\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s}|c_{W},N_{h},N_{s})\,|\vec{q}_{\text{opt}}|^{2},$ (27)

Taking into account the particle multiplicity, the joint probability
distribution $p({\cal Q}_{\kappa,h},{\cal Q}_{\kappa,s}|c_{W},N_{h},N_{s})$
becomes

$\displaystyle p({\cal Q}_{\kappa,h},{\cal Q}_{\kappa,s}|c_{W},N_{h},N_{s})$
$\displaystyle=\left[p({\cal Q}_{\kappa,h}|N_{h},\bar{d}\to
q_{\text{hard}})\,p(\bar{d}\to q_{\text{hard}}|c_{W})+p({\cal
Q}_{\kappa,h}|N_{h},u\to q_{\text{hard}})\,p(u\to
q_{\text{hard}}|c_{W})\right]$ $\displaystyle\times\left[p({\cal
Q}_{\kappa,s}|N_{s},\bar{d}\to q_{\text{soft}})\,p(\bar{d}\to
q_{\text{soft}}|c_{W})+p({\cal Q}_{\kappa,s}|N_{s},u\to
q_{\text{soft}})\,p(u\to q_{\text{soft}}|c_{W})\right],$ (28)

where $p({\cal Q}|N,\bar{d})$ and $p({\cal Q}|N,u)$ take the Gaussian forms
from Eqs. (15) and (16).

Figure 1: Spin analyzing power
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}$ as a function of the
particle multiplicity $N_{s}$, assuming $N_{h}+N_{s}=20$. We present the
results for different values of $\kappa$, taking $\sigma_{z}^{2}N^{2}=1/4$
(left panel) and $\sigma_{z}^{2}N^{2}=1$ (right panel).

Figure 2: Square root of the mean square optimal direction as a function of
multiplicity and helicity angle (upper left), jet charge of the hard jet and
helicity angle (middle left), jet charge of the soft jet and helicity angle
(bottom left), jet charge of the hard jet and multiplicity (upper right), jet
charge of the soft jet and multiplicity (middle right), and jet charge of the
hard and soft jets (bottom right). We assume $\kappa=0.3$ and
$\sigma_{z}^{2}N^{2}=1$.

The integrand in Eq. (27) is a complicated function of ${\cal Q}_{\kappa,h}$,
${\cal Q}_{\kappa,s}$ and $c_{W}$, so we perform the numerical integration for
a fixed particle multiplicity $N_{s}$, assuming $N_{s}+N_{h}=20$, and
different values of $\kappa$. To illustrate the dependence on the width, we
will present results with $\sigma_{z}^{2}N^{2}=(1/4,~{}1)$. Our results are
presented in Fig. 1, which shows that
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}$ presents a U-shape profile
due to the fixed $N_{s}+N_{h}$. The spin analyzing power increases for small
$N_{s}$ with large $N_{h}$, as well as large $N_{s}$ with small $N_{h}$. In
addition, we note that the spin analyzing power slightly increases for smaller
$\kappa$. This is in agreement with the results reported in Ref. Kang:2023ptt
, indicating an increase in quark discrimination power as $\kappa$ approaches
zero.

To further investigate a possible enhancement of the spin analyzing power, we
explore two-dimensional correlations between the observables: multiplicity of
the soft jet $N_{s}$, the charges of the hard and soft jets $({\cal
Q}_{\kappa,h},{\cal Q}_{\kappa,s})$, and helicity angle $c_{W}$. For this
purpose, we defined the average of the squared-vector length over a region,
${\cal R}$,

$\displaystyle\langle|\vec{q}_{\text{opt}}|^{2}\rangle_{{\cal
R}}=\frac{\int_{{\cal R}}\,dc_{W}\,d{\cal Q}_{\kappa,h}\,d{\cal
Q}_{\kappa,s}\,p(c_{W})\,p({\cal Q}_{\kappa,h},{\cal
Q}_{\kappa,s}|c_{W},N_{h},N_{s})\,|\vec{q}_{\text{opt}}|^{2}}{\int_{{\cal
R}}\,dc_{W}\,d{\cal Q}_{\kappa,h}\,d{\cal Q}_{\kappa,s}\,p(c_{W})\,p({\cal
Q}_{\kappa,h},{\cal Q}_{\kappa,s}|c_{W},N_{h},N_{s})},$ (29)

where $\cal R$ denotes each bin of the two-dimensional distribution. The
corresponding results are presented in Fig. 2. We observe that the spin
analyzing power increases for large helicity angles, getting close to unity as
$c_{W}$ approaches 1. In this limit, the down quark rarely decays collinear to
the $W$ boson in the top quark rest frame because $f_{R}\simeq 0$ in Eq. (23).
Hence, it is possible to obtain the full spin analyzing power in this
kinematic regime. Furthermore, there is a significant increase in spin
analyzing power for particular jet charge values. More concretely, we observe
an enhancement for large and positive ${\cal Q}_{\kappa,h}$ (large and
negative ${\cal Q}_{\kappa,s}$). This phenomenological behavior was
anticipated in the discussion after Eq. (14). These features can be easily
incorporated as a part of event selection at the cost of statistics, taking
advantage of the large branching fraction for the hadronic top quark final
state.

Figure 3: In the top panel, we present the average of the spin analyzing power $\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}$ as a function of the helicity angle $c_{W}$ (left), jet charge for the soft jet ${\cal Q}_{\kappa,s}$ (middle), and hard jet ${\cal Q}_{\kappa,h}$ (right). The results are shown for different multiplicities of the soft jet $N_{s}=3$ (black), $N_{s}=5$ (blue), and $N_{s}=7$ (red), assuming $N_{s}+N_{h}=20$. In the bottom panel, we show the average of the spin analyzing power for distinct efficiency cuts on $c_{W}$ (left), ${\cal Q}_{\kappa,\bar{d}\to q_{\text{soft}}}^{\text{max}}$ (middle), and ${\cal Q}_{\kappa,\bar{d}\to q_{\text{hard}}}^{\text{min}}$ (right). We assume $\kappa=0.3$ and $\sigma_{z}^{2}N^{2}=1$. | $\text{Eff}=100\%$ | $\text{Eff}(c_{W}^{\rm min})=50\%$ | $\text{Eff}(c_{W}^{\rm min})=20\%$ | $\text{Eff}({\cal Q}_{\kappa,s}^{\rm max})=50\%$ | $\text{Eff}({\cal Q}_{\kappa,s}^{\rm max})=20\%$
---|---|---|---|---|---
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}_{N_{s}=3}$ |  0.646 |  0.649 |  0.695 |  0.666 |  0.683
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}_{N_{s}=5}$ |  0.644 |  0.647 |  0.694 |  0.659 |  0.672
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}_{N_{s}=7}$ |  0.644 |  0.647 |  0.693 |  0.655 |  0.667
Table 1: Spin analyzing power
$\sqrt{\langle|\vec{q}_{\text{opt}}|^{2}\rangle}$ for different efficiencies
and soft jet multiplicities $N_{s}=(3,5,7)$.

In Fig. 3, we illustrate the magnitude of this improvement. The top panel
displays the spin analyzing power as a function of helicity angle, and jet
charges for the soft and hard jets. The results are shown for distinct soft
jet multiplicities $N_{s}=(3,5,7)$, assuming $N_{s}+N_{h}=20$. While the
average of the spin analyzing power as a function of the helicity angle does
not show significant changes for different $N_{s}$ values (and the changes for
${\cal Q}_{\kappa,h}$ version of this plot are only mild), the average of the
spin analyzing power as a function of the jet charge ${\cal Q}_{\kappa,s}$
shows relevant changes for different $N_{s}$. Based on the profiles of these
distributions, we define minimal or maximal thresholds in the phase space to
enhance the spin analyzing power. This is presented in terms of the efficiency
of the corresponding selection, shown in the bottom panel of the same figure
and illustrated in Table 1 for particular efficiency values. We observe that
by implementing a minimal threshold on the helicity angle, $c_{W}>c_{W}^{\rm
min}$, it is possible to increase the spin analyzing power from 0.64 by 1.4%
(8.6%) for 0.5 (0.2) efficiency. Similarly, we examine the potential
enhancement arising from the jet charge distributions. In particular, for
${\cal Q}_{\kappa,\bar{d}\to\rm q_{soft}}<{\cal Q}_{\kappa,\bar{d}\to\rm
q_{soft}}^{\rm max}$, the possible increase can reach up to approximately 4.0%
(6.7%) for 0.5 (0.2) efficiency. Hence, considering the studied observables,
the major gains for the hadronic top quark polarimetry will come from a
combination of $c_{W}$ and ${\cal Q}_{\kappa,s}$ dependencies.

##  IV Conclusion

The measurement of top quark polarization provides a unique and powerful tool
to advance precision physics and explore new physics beyond the Standard
Model. While its leptonic decay serves as a clean proxy for the top quark
polarimetry, it comes with significant statistical limitations. In contrast,
the sizable hadronic decays of the top quark, despite their complexity, offer
a promising avenue for polarization studies Tweedie:2014yda ;
PhysRevD.109.115023 ; Dong:2024xsg ; CMS:2024vqh .

In this paper, we investigated the relevance of global jet dynamics – arising
from kinematics, jet charges, and particle multiplicity – for enhancing the
sensitivity to hadronic top quark polarimetry. By applying the general
formalism presented in Ref. Dong:2024xsg , we provided analytic parametric
estimates of the improvement in the spin analyzing power for hadronic top
quarks. Our analysis demonstrated that the considered observables can lead to
substantial improvements in specific regions of phase space. In particular, we
observed that the helicity angle and jet charge for the soft jet yield sizable
effects, underscoring their importance in hadronic top quark polarimetry.
These analytical derivations provide deeper insights into the sources of
improvement in hadronic top quark polarimetry, building on the foundation laid
in Ref. Dong:2024xsg .

Acknowledgments. The authors thank the organizers for the Mitchell Conference
on Collider, Dark Matter, and Neutrino Physics held at Texas A&M University,
where the inspiration for this project originated. D.G. thanks the group at
the IPPP-Durham University for hosting him during the final stages of this
project. DG and AN thank the U.S. Department of Energy for the financial
support, under grant number DE-SC 0016013. A.L. was supported in part by the
UC Southern California Hub, with funding from the UC National Laboratories
division of the University of California Office of the President. K.K. is
supported by US DOE DE-SC0024407 and Z.D. is supported in part by US DOE DE-
SC0024673 and by College of Liberal Arts and Sciences Research Fund at the
University of Kansas. K.K. would like to thank the Aspen Center for Physics
and the organizers of Summer 2024 workshop, “Fundamental Physics in the Era of
Big Data and Machine Learning” (supported by National Science Foundation grant
PHY-2210452) for hospitality during the completion of this manuscript.

##  Appendix A Enhancing Spin Analyzing Power with Jet Charge: a Simple Proof

In this appendix, we provide a simple proof that the spin analyzing power is
always improved by incorporating jet charge information. The spin analyzing
power can be examined in terms of the length of the vector

$\displaystyle\vec{q}_{\text{opt}}$ $\displaystyle=p(\bar{d}\to
q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal
Q}_{\kappa,s},N_{s})\,\hat{q}_{\text{hard}}+p(\bar{d}\to
q_{\text{soft}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal
Q}_{\kappa,s},N_{s})\,\hat{q}_{\text{soft}}\,.$ (30)

Then, $|\vec{q}_{\text{opt}}|^{2}$ becomes

$\displaystyle|\vec{q}_{\text{opt}}|^{2}$
$\displaystyle=1-2x(1-x)(1-\hat{q}_{\text{hard}}\cdot\hat{q}_{\text{soft}})\,,$
(31)

where $x\equiv p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})$. For a fixed value of the dot
product $\hat{q}_{\text{hard}}\cdot\hat{q}_{\text{soft}}$, this length
increases if the product $x(1-x)$ decreases. With this in mind, we can analyze
how the inclusion of jet charge improves the spin resolving power.

Assuming that the measured value of $c_{W}$ is such that $p(\bar{d}\to
q_{\text{hard}}|c_{W})<1/2$, we will argue that $p(\bar{d}\to
q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal
Q}_{\kappa,s},N_{s})<p(\bar{d}\to q_{\text{hard}}|c_{W})$, and correspondingly
that the spin analyzing power has improved by measuring the jet charge
information. Consider the first factor of $p(\bar{d}\to
q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})$ in
Eq. (14)

$\displaystyle\frac{1}{p(\bar{d}\to q_{\text{hard}}|c_{W})+\frac{p({\cal
Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})}{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}},N_{h})}\,p(u\to q_{\text{hard}}|c_{W})}\,.$ (32)

Necessarily, we have the sum rule, $p(\bar{d}\to q_{\text{hard}}|c_{W})+p(u\to
q_{\text{hard}}|c_{W})=1$. Because we assume that $p(\bar{d}\to
q_{\text{hard}}|c_{W})<1/2$, most of the time the harder jet is $u$, and so on
average the ratio

$\displaystyle\frac{p({\cal Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})}{p({\cal
Q}_{\kappa,h}|\bar{d}\to q_{\text{hard}},N_{h})}>1\,.$ (33)

Therefore, the first factor is less than 1:

$\displaystyle\frac{1}{p(\bar{d}\to q_{\text{hard}}|c_{W})+\frac{p({\cal
Q}_{\kappa,h}|u\to q_{\text{hard}},N_{h})}{p({\cal Q}_{\kappa,h}|\bar{d}\to
q_{\text{hard}},N_{h})}\,p(u\to q_{\text{hard}}|c_{W})}<1\,.$ (34)

A similar argument holds for the second factor in Eq. (14), and we have

$\displaystyle\frac{1}{\frac{p({\cal Q}_{\kappa,s}|\bar{d}\to
q_{\text{soft}},N_{s})}{p({\cal Q}_{\kappa,s}|u\to
q_{\text{soft}},N_{s})}\,p(\bar{d}\to q_{\text{soft}}|c_{W})+p(u\to
q_{\text{soft}}|c_{W})}<1\,.$ (35)

Thus, we necessarily find that $p(\bar{d}\to q_{\text{hard}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})<p(\bar{d}\to
q_{\text{hard}}|c_{W})$. Similarly, it is also true that $p(\bar{d}\to
q_{\text{soft}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal
Q}_{\kappa,s},N_{s})>p(\bar{d}\to q_{\text{soft}}|c_{W})$ for the same value
of $c_{W}$.

Now, $x(1-x)$ is maximized where $x=1/2$, but we assumed that we started away
from the maximum and with the inclusion of jet charge moved further from the
maximum because the difference $p(\bar{d}\to q_{\text{soft}}|c_{W},{\cal
Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})-p(\bar{d}\to
q_{\text{hard}}|c_{W},{\cal Q}_{\kappa,h},N_{h},{\cal Q}_{\kappa,s},N_{s})$ is
larger than $p(\bar{d}\to q_{\text{soft}}|c_{W})-p(\bar{d}\to
q_{\text{hard}}|c_{W})$. Therefore, the spin analyzing power must be improved
by measuring jet charge information.

## References

  * [1] Werner Bernreuther and Arnd Brandenburg.  Tracing CP violation in the production of top quark pairs by multiple TeV proton proton collisions.  Phys. Rev. D, 49:4481–4492, 1994.
  * [2] Masato Arai, Nobuchika Okada, Karel Smolek, and Vladislav Simak.  Top quark spin correlations in the Randall-Sundrum model at the CERN Large Hadron Collider.  Phys. Rev. D, 75:095008, 2007.
  * [3] Maxim Perelstein and Andreas Weiler.  Polarized Tops from Stop Decays at the LHC.  JHEP, 03:141, 2009.
  * [4] David Krohn, Tao Liu, Jessie Shelton, and Lian-Tao Wang.  A Polarized View of the Top Asymmetry.  Phys. Rev. D, 84:074034, 2011.
  * [5] Vernon Barger, Wai-Yee Keung, and Brian Yencho.  Azimuthal Correlations in Top Pair Decays and The Effects of New Heavy Scalars.  Phys. Rev. D, 85:034016, 2012.
  * [6] Matthew Baumgart and Brock Tweedie.  Discriminating Top-Antitop Resonances using Azimuthal Decay Correlations.  JHEP, 09:049, 2011.
  * [7] Matthew R. Buckley and Dorival Gonçalves.  Boosting the Direct CP Measurement of the Higgs-Top Coupling.  Phys. Rev. Lett., 116(9):091801, 2016.
  * [8] Matthew R. Buckley and Dorival Gonçalves.  Constraining the Strength and CP Structure of Dark Production at the LHC: the Associated Top-Pair Channel.  Phys. Rev. D, 93(3):034003, 2016.
  * [9] Dorival Gonçalves and David Lopez-Val.  Pseudoscalar searches with dileptonic tops and jet substructure.  Phys. Rev. D, 94(9):095005, 2016.
  * [10] Dorival Gonçalves, Kyoungchul Kong, and Jeong Han Kim.  Probing the top-Higgs Yukawa CP structure in dileptonic $t\overline{t}h$ with M2-assisted reconstruction.  JHEP, 06:079, 2018.
  * [11] Vernon Barger, Kaoru Hagiwara, and Ya-Juan Zheng.  Probing the top Yukawa coupling at the LHC via associated production of single top and Higgs.  JHEP, 09:101, 2020.
  * [12] Dorival Gonçalves, Jeong Han Kim, Kyoungchul Kong, and Yongcheng Wu.  Direct Higgs-top CP-phase measurement with $t\overline{t}h$ at the 14 TeV LHC and 100 TeV FCC.  JHEP, 01:158, 2022.
  * [13] Rahool Kumar Barman, Dorival Gonçalves, and Felix Kling.  Machine learning the Higgs boson-top quark CP phase.  Phys. Rev. D, 105(3):035023, 2022.
  * [14] Roshan Mammen Abraham and Dorival Gonçalves.  Boosting new physics searches in $t{\bar{t}}Z$ and tZj production with angular moments.  Eur. Phys. J. C, 83(10):965, 2023.
  * [15] Fabio Maltoni, Claudio Severi, Simone Tentori, and Eleni Vryonidou.  Quantum detection of new physics in top-quark pair production at the LHC.  JHEP, 03:099, 2024.
  * [16] Gregory Mahlon and Stephen J. Parke.  Angular correlations in top quark pair production and decay at hadron colliders.  Phys. Rev. D, 53:4886–4896, 1996.
  * [17] Gregory Mahlon and Stephen J. Parke.  Spin Correlation Effects in Top Quark Pair Production at the LHC.  Phys. Rev. D, 81:074024, 2010.
  * [18] T. Aaltonen et al.  Measurement of $t\bar{t}$ Spin Correlation in $p\bar{p}$ Collisions Using the CDF II Detector at the Tevatron.  Phys. Rev. D, 83:031104, 2011.
  * [19] Victor Mukhamedovich Abazov et al.  Measurement of spin correlation in $t\bar{t}$ production using dilepton final states.  Phys. Lett. B, 702:16–23, 2011.
  * [20] Morad Aaboud et al.  Measurements of top-quark pair spin correlations in the $e\mu$ channel at $\sqrt{s}=13$ TeV using $pp$ collisions in the ATLAS detector.  Eur. Phys. J. C, 80(8):754, 2020.
  * [21] Albert M Sirunyan et al.  Measurement of the top quark polarization and $\mathrm{t\bar{t}}$ spin correlations using dilepton final states in proton-proton collisions at $\sqrt{s}=$ 13 TeV.  Phys. Rev. D, 100(7):072002, 2019.
  * [22] J. A. Aguilar-Saavedra, M. C. N. Fiolhais, P. Martín-Ramiro, J. M. Moreno, and A. Onofre.  A template method to measure the $t{\bar{t}}$ polarisation.  Eur. Phys. J. C, 82(2):134, 2022.
  * [23] J. A. Aguilar-Saavedra.  Crafting polarizations for top, W, and Z.  Phys. Rev. D, 106(11):115021, 2022.
  * [24] Yoav Afik and Juan Ramón Muñoz de Nova.  Entanglement and quantum tomography with top quarks at the LHC.  Eur. Phys. J. Plus, 136(9):907, 2021.
  * [25] M. Fabbrichesi, R. Floreanini, and G. Panizzo.  Testing Bell Inequalities at the LHC with Top-Quark Pairs.  Phys. Rev. Lett., 127(16):161801, 2021.
  * [26] Claudio Severi, Cristian Degli Esposti Boschi, Fabio Maltoni, and Maximiliano Sioli.  Quantum tops at the LHC: from entanglement to Bell inequalities.  Eur. Phys. J. C, 82(4):285, 2022.
  * [27] J. A. Aguilar-Saavedra and J. A. Casas.  Improved tests of entanglement and Bell inequalities with LHC tops.  Eur. Phys. J. C, 82(8):666, 2022.
  * [28] Yoav Afik and Juan Ramón Muñoz de Nova.  Quantum information with top quarks in QCD.  Quantum, 6:820, 2022.
  * [29] Yoav Afik and Juan Ramón Muñoz de Nova.  Quantum Discord and Steering in Top Quarks at the LHC.  Phys. Rev. Lett., 130(22):221801, 2023.
  * [30] Claudio Severi and Eleni Vryonidou.  Quantum entanglement and top spin correlations in SMEFT at higher orders.  JHEP, 01:148, 2023.
  * [31] Rafael Aoude, Eric Madge, Fabio Maltoni, and Luca Mantani.  Quantum SMEFT tomography: Top quark pair production at the LHC.  Phys. Rev. D, 106(5):055007, 2022.
  * [32] Zhongtian Dong, Dorival Gonçalves, Kyoungchul Kong, and Alberto Navarro.  Entanglement and bell inequalities with boosted $t\overline{t}$.  Phys. Rev. D, 109:115023, Jun 2024.
  * [33] Tao Han, Matthew Low, and Tong Arthur Wu.  Quantum Entanglement and Bell Inequality Violation in Semi-Leptonic Top Decays.  10 2023.
  * [34] Kun Cheng, Tao Han, and Matthew Low.  Optimizing fictitious states for Bell inequality violation in bipartite qubit systems with applications to the $t\bar{t}$ system.  Phys. Rev. D, 109(11):116005, 2024.
  * [35] Georges Aad et al.  Observation of quantum entanglement in top-quark pairs using the ATLAS detector.  11 2023.
  * [36] Alan J. Barr, Marco Fabbrichesi, Roberto Floreanini, Emidio Gabrielli, and Luca Marzola.  Quantum entanglement and Bell inequality violation at colliders.  2 2024.
  * [37] Probing entanglement in top quark production with the CMS detector.  Technical report, CERN, Geneva, 2024.
  * [38] Measurements of polarization, spin correlations, and entanglement in top quark pairs using lepton+jets events from pp collisions at $\sqrt{s}=13~{}\mathrm{TeV}$.  CMS Collaboration, CMS-PAS-TOP-23-007, 2024.
  * [39] Zhongtian Dong, Dorival Gonçalves, Kyoungchul Kong, Andrew J. Larkoski, and Alberto Navarro.  Hadronic Top Quark Polarimetry with ParticleNet.  7 2024.
  * [40] Kun Cheng, Tao Han, and Matthew Low.  Optimizing Entanglement and Bell Inequality Violation in Top Anti-Top Events.  7 2024.
  * [41] Werner Bernreuther and Zong-Guo Si.  Distributions and correlations for top quark pair production and decay at the Tevatron and LHC.  Nucl. Phys. B, 837:90–121, 2010.
  * [42] M. Jezabek.  Top quark physics.  Nucl. Phys. B Proc. Suppl., 37(2):197, 1994.
  * [43] Brock Tweedie.  Better Hadronic Top Quark Polarimetry.  Phys. Rev. D, 90(9):094010, 2014.
  * [44] R. D. Field and R. P. Feynman.  A Parametrization of the Properties of Quark Jets.  Nucl. Phys. B, 136:1, 1978.
  * [45] David Krohn, Matthew D. Schwartz, Tongyan Lin, and Wouter J. Waalewijn.  Jet Charge at the LHC.  Phys. Rev. Lett., 110(21):212001, 2013.
  * [46] Katherine Fraser and Matthew D. Schwartz.  Jet Charge and Machine Learning.  JHEP, 10:093, 2018.
  * [47] Zhong-Bo Kang, Andrew J. Larkoski, and Jinghong Yang.  Towards a Nonperturbative Formulation of the Jet Charge.  Phys. Rev. Lett., 130(15):151901, 2023.
  * [48] R. L. Workman et al.  Review of Particle Physics.  PTEP, 2022:083C01, 2022.
  * [49] M. Aaboud et al.  Search for the Decay of the Higgs Boson to Charm Quarks with the ATLAS Experiment.  Phys. Rev. Lett., 120(21):211802, 2018.
  * [50] Performance summary of AK4 jet charm tagging with the CMS Run2 Legacy dataset.  2023\.
  * [51] Zhong-Bo Kang, Robert Kao, and Andrew J. Larkoski.  Multiplicity scaling of fragmentation function.  Phys. Rev. D, 109(5):054039, 2024.
  * [52] Z. Bern et al.  Left-Handed W Bosons at the LHC.  Phys. Rev. D, 84:034008, 2011.